Jean-François Marckert

Grégory Miermont

The CRT is the scaling limit of unordered binary trees

We prove that a uniform, rooted unordered binary tree with n vertices has the Brownian continuum random tree as its scaling limit for the Gromov-Hausdorff topology. The limit is thus, up to a constant factor, the same as that of uniform plane trees or labeled trees. Our analysis rests on a combinatorial and probabilistic study of appropriate trimming procedures of trees.

1 This can also be obtained in the framework of Galton-Watson trees by choosing the particular Poisson offspring distribution

Introduction

The Brownian Continuum Random Tree (CRT), introduced by Aldous [START_REF] Aldous | The continuum random tree[END_REF], is a natural object that arises in various situations in Probability Theory. It is known to be the universal scaling limit for conditioned critical Galton-Watson trees with finite variance offspring distribution [START_REF] Aldous | The continuum random tree[END_REF][START_REF] Marckert | The depth first processes of Galton-Watson trees converge to the same Brownian excursion[END_REF][START_REF] Duquesne | A limit theorem for the contour process of conditioned Galton-Watson trees[END_REF], or of random labeled trees on n vertices (Cayley trees) [START_REF] Aldous | The continuum random tree[END_REF][START_REF] Camarri | Limit distributions and random trees derived from the birthday problem with unequal probabilities[END_REF][START_REF] Aldous | The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin's local time identity[END_REF].

Several distinct proofs for the convergence of discrete trees towards the CRT exist in the literature, taking advantage of the specific aspects of the considered models, which in passing yield various equivalent constructions of the CRT. For instance, a specific 1 proof of the convergence of Cayley trees rests on the fact that a uniform Cayley tree is a uniform spanning tree of the complete graph, which can be constructed via the Aldous-Broder algorithm and leads to a limiting "stick-breaking construction" of the CRT. On the other hand, the convergence of conditioned Galton-Watson trees to the CRT can be obtained by appropriate encodings of trees by random walks. In [START_REF] Aldous | The continuum random tree. II. An overview[END_REF], Aldous conjectures that many other models of trees, for which it is harder to have a good probabilistic understanding, also have the CRT as a continuum limit.

In this article, we will focus on one of these models, namely, the family of rooted binary unordered trees, considered as graph-theoretic trees without planar or labeled structure. The main goal of this paper is to prove that a uniformly chosen rooted binary unordered tree with n leaves converges, after renormalization of distances by √ n, to a constant multiple of the CRT.

From a combinatorial perspective, models of plane trees are very close to the case studied here: their generating functions' singularities are of the same type (a square-root singularity), and heuristically, combinatorial families with generating functions bearing the same singularity type should have a similar continuous limit. Nevertheless, the probabilistic methods developed for plane or labeled trees are not valid anymore for unordered trees, and another method has to be developed to derive their scaling limits. The present work is a step in this direction, for the particular case of binary, rooted, unordered trees.

We mention that in a very recent paper, another model of rooted unlabeled, unordered trees has been investigated by Drmota & Gittenberger [START_REF] Drmota | The shape of unlabeled rooted random trees[END_REF]: this is the model of Pólya trees where there is no restriction on the authorized degrees. They compute the limiting height and limiting profile of these trees (with n nodes, when n → +∞) and show that, up to constant factors, they respectively converge to the maximum of the Brownian excursion, and the local time of the Brownian excursion (which is similar to the case of binary plane trees). Even if this result is of a different kind from ours, and does not imply the convergence of the rescaled tree to the CRT for the Gromov-Hausdorff topology, it gives an important indication that it should be the case.

Plane trees

Let us recall the standard definition of plane trees (see Figures 1,2 and 3). Let U = n≥0 N n be the set of words with integer letters, where N = {1, 2, . . .} and N 0 = {∅}. For u ∈ U, let |u| be its length, and we generally note u = u 1 . . . u n with n = |u|. The concatenation of the words u and v is written uv, and we write u v if u is a prefix of v, meaning that there exists w ∈ U such that uw = v. This defines a partial order on U. If A ⊂ U and u ∈ U, we let uA = {uv : v ∈ A} .

If u, v ∈ U, we let u ∧ v be the longest common prefix to u and v. The set U is endowed with the lexicographical order ≤: we have u ≤ v if u v or if u ∧ v is a strict prefix of u and v such that u |u∧v|+1 < v |u∧v|+1 (on N). The order (U, ≤) is total.

Definition 1 A rooted plane tree is a finite subset Ø ⊂ U containing ∅, such that if ui ∈ Ø with u ∈ U and i ∈ N, then

• uj ∈ Ø for 1 ≤ j ≤ i, and

• u ∈ Ø.
(See Figure 2). The elements of Ø are called vertices, and ∅ is called the root. A vertex ui ∈ Ø, with u ∈ U and i ∈ N, is called a child of u. Their number is denoted by c u (Ø) = sup{i ≥ 1 : ui ∈ Ø} ∈ Z + = {0, 1, 2, . . .}. The notion of brothers, ancestors, descendants are induced by that of child as in the standard life. The length |u| is equally called height of u.

In a plane tree Ø, the subtree of Ø rooted at u ∈ Ø is the plane tree Ø u = {v ∈ U : uv ∈ Ø}.

A plane tree has a representation as a plane graph, where each vertex is linked by an edge to its children, which are ordered from left to right in lexicographical order (as done on Figure 2). We say that a rooted plane tree Ø is binary if c u (Ø) ∈ {0, 2} for every u ∈ Ø. The vertices having no children are called the leaves the set of which is denoted by L(Ø), while I(Ø) = Ø \ L(Ø) denotes the set of internal vertices. In turn, the internal vertices can be partitioned into three sets I(Ø) = I 0 (Ø) ⊔ I 1 (Ø) ⊔ I 2 (Ø) ,

where for i ∈ {0, 1, 2}, I i (Ø) is the set of internal vertices having i children being themselves internal vertices (see Figure 2). We let S(Ø) = L(Ø) \ {u1, u2 : u ∈ I 0 (Ø)} , the set of leaves that are children of the vertices of I 1 (Ø). These leaves will be called "skeleton leaves" of Ø. We let |Ø| denote the number of leaves of Ø. We let T be the set of binary rooted plane trees, and T n = {Ø : |Ø| = n} ⊂ T be the subset of those having n leaves (hence 2n -1 vertices and 2n -2 edges). It is a simple exercise to enumerate rooted binary plane trees via their generating functions. Namely, we have

#T n = 1 n 2n -2 n -1 ,
the (n -1)-th Catalan number, which also counts numerous families of combinatorial objects. Note the equivalent

#T n ∼ n→∞ 4 n-1 √ πn 3/2 . (1)

Unordered trees

A rooted, binary unordered tree is a rooted, binary plane tree in which the planar order has been "forgotten" (but not the root). Two rooted binary plane trees Ø and Ø ′ are equivalent if Ø ′ can be obtained from Ø by exchanging the order of the various children of Ø, while preserving the rooted genealogical structure: ancestors and brothers remain as such in the process. Formally, consider Ø ∈ T and a family of permutations σ = (σ u , u ∈ I(Ø)) of the set {1, 2}, indexed by the internal vertices of Ø. For u = u 1 . . . u n ∈ Ø, we let

σ(u) = σ ∅ (u 1 )σ u 1 (u 2 ) . . . σ u 1 ...u n-1 (u n ) ,
and σ(∅) = ∅. Now for Ø ′ ∈ T we say that Ø ∼ Ø ′ if there exists such a σ with Ø ′ = σ(Ø) .

Note that there are at most 2 |Ø|-1 elements in the class of Ø, since #I(Ø) = |Ø| -1.

Ø Ø ′

∅ ∅ Figure 3: Two plane trees Ø and Ø ′ that are equal as trees.

Definition 2

The quotient set T = T/ ∼ is called the set of (binary, rooted, unordered) trees. projection. We also let T n = T n / ∼ be those trees which have n leaves, and set |t| = n for t ∈ T n .

It will be convenient to fix once and for all a section s : T → T of the canonical projection T → T, so s(t) is a choice among all the planar representatives of t. In this way, there will be a non-ambiguous way to consider a vertex of an unordered tree, the number of its children, one of its edges, and so on.

The numbers (#T n , n ≥ 1) are the Etherington-Wedderburn numbers, the first ones are 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, 451, ... (referred to as A001190 in Sloane's On-Line Encyclopedia of Integer Sequences). No closed formula is available, but the asymptotic enumeration of elements of T has been studied by Otter [START_REF] Otter | The number of trees[END_REF], using the properties of the generating function for trees 2 . Let T(z) = n≥1 #T n z n be the generating function for rooted binary trees, counted according to their number of leaves. A decomposition of a tree at its root into a multiset of two subtrees yields the formula

T(z) = z + T(z 2 ) + T(z) 2 2 . ( 2
)
One shows that the radius of convergence ̺ of T satisfies

T(̺) = 1 . (3)
A detailed analysis of the recursive equation ( 2) entails

#T n ∼ n→∞ c 2 √ πn 3/2 ̺ n , (4)
where

c := 2̺ + 2̺ 2 T ′ (̺ 2 ) = 1.1300337 . . . , (5)
see [START_REF] Flajolet | Analytic Combinatorics[END_REF]Note VII.22,p.477]. The equivalent ( 4) is similar to [START_REF] Aldous | The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin's local time identity[END_REF], which hints at the similarity of structure of large trees with large planar trees.

Important notice. From this section onwards, we will only be interested in rooted, binary, plane or unordered trees, and the words "rooted, binary", will always be implicit when dealing with the two combinatorial families of trees we have just defined. To avoid confusion, from now on,

• a plane tree stands for a rooted, binary plane tree,

• while a tree is a rooted, binary, unordered tree.

Gromov-Hausdorff topology

It is natural to consider trees (plane or not) as finite metric spaces, by endowing the set of their vertices with the usual graph distance. Formally, for Ø ∈ T and u, v ∈ Ø, we let

d Ø (u, v) = |u| + |v| -2|u ∧ v| .
In the sequel, we will often improperly identify Ø with the metric space (Ø, d Ø ), or even with its isometry class, i.e. the collection of metric spaces that are isometric to it. It is straightforward that (Ø, d Ø ) and (Ø ′ , d Ø ′ ) are isometric spaces whenever Ø ∼ Ø ′ , so that a tree t will also denote the isometry class of the metric space (s(t), d s(t) ). We will adopt the notation a(X, d) = (X, ad) for (X, d) a metric space (identified with its isometry class) and a > 0. Therefore, the notation aØ, at will stand for the metric spaces Ø, t with distances multiplied by a.

There is a common way to define a topology (even a metric) on spaces of metric spaces. Such topologies have been developed in Geometry for the last 30 years, following the ideas of Gromov [START_REF] Gromov | Metric structures for Riemannian and non-Riemannian spaces[END_REF]. Their use in Probability in the context of random real trees has been popularized by Evans and his coauthors [START_REF] Evans | Snakes and spiders: Brownian motion on R-trees[END_REF][START_REF] Evans | Rayleigh processes, real trees, and root growth with re-grafting[END_REF][START_REF] Evans | Subtree prune and regraft: a reversible real tree-valued Markov process[END_REF], and has been applied successfully in various situations, for instance by Duquesne & Le Gall [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF], Le Gall [START_REF] Gall | The topological structure of scaling limits of large planar maps[END_REF]. We let M be the set of isometry classes of compact metric spaces, which we endow with the Gromov-Hausdorff distance defined by

d GH ((X, d), (X ′ , d ′ )) = inf φ,φ ′ δ H (φ(X), φ ′ (X ′ )) , (6)
the infimum being taken over the set of isometric embeddings φ, φ ′ from (X, d), (X ′ , d ′ ) into a common metric space (Z, δ), and where δ H is the usual Hausdorff distance between compact subsets of Z. The reader can consult [START_REF] Burago | A course in metric geometry[END_REF]Chapter 7] for basic properties of this distance, which turns M into a Polish metric space (this is a straightforward extension of the proof of [START_REF] Evans | Rayleigh processes, real trees, and root growth with re-grafting[END_REF]Theorem 1]).

Of particular importance for our purposes is the subset T ⊂ M of R-trees, i.e. of compact spaces (T , d) such that for every pair of points x, y ∈ T ,

• there exists an isometry ϕ x,y : [0, d(x, y)] → T with ϕ x,y (0) = x and ϕ x,y (d(x, y)) = y.

• for every continuous injective q : [0, 1] → T with q(0) = x and q(1) = y, it holds that

q([0, 1]) = ϕ x,y ([0, d(x, y)]).
The first property says that (T , d) is a geodesic space, and the second is a "tree property" that there is a unique way to travel between two points without backtracking. The set T is closed in M . We should mention that when dealing with the convergence of plane trees, a natural topology is inherited from the uniform distance between (rescaled) contour processes, as defined later in Section 4.4. This topology is stronger than the the Gromov-Hausdorff topology, since contour processes encode also the lexicographical order of the vertices. For unordered trees, the Gromov-Hausdorff distance is very natural since it is by essence a distance on metric spaces, which ignores any extra structure, like planarity.

Finally, we stress that there is a variant of the Gromov-Hausdorff distance on pointed metric spaces [START_REF] Evans | Rayleigh processes, real trees, and root growth with re-grafting[END_REF], which would be natural in our context since we are interested in rooted trees. It is straightforward to check that the results to come remain valid in this setting, by taking the root vertex as the distinguished point.

The Brownian continuum random tree

In order to state our main result, it remains to briefly describe the CRT, which arises as the scaling limit of plane trees, as we now recall.

Let E be the set of continuous functions f defined on an interval [0, σ f ], that are non-negative, and satisfy f (0) = f (σ f ) = 0. It is a complete space when endowed with the distance

D(f, g) = sup t≥0 |f (t ∧ σ f ) -g(t ∧ σ g )| .
With every f ∈ E, we can associate an R-tree, following [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF]. Define a function

d f on [0, σ f ] 2 by d f (s, t) = f s + f t -2 fs,t ,
where by definition fs,t = inf s∧t≤u≤s∨t f u .

It is easy to see that d f is a pseudo-distance on [0, σ f ]. It is not a distance as it does not separate points, so we let s ≡ f t if d f (s, t) = 0, defining an equivalence relation on [0, σ f ] (see Figure 4).

0 σ f s s ′ t f (s) f (t) f (s, t)
Figure 4: Graph of a function f from E. In this example s ≡ f s ′ and the distance

d f (s, t) = d f (s ′ , t) = f (s) + f (t) -2 f (s, t).
The quotient space T f := [0, σ f ]/ ≡ f is endowed with the distance induced by d f , which we still call d f . The canonical projection [0, σ f ] → T f being obviously continuous, the target space is compact. The (isometry class of the) space T f is thus an element of M , and turns out to be an element of T . The space T f is naturally rooted, i.e. comes with a distinguished point ρ, which is the ≡ f -equivalence class of 0. Finally, the mapping f → T f is continuous from (E, D) to (T , d GH ). Now, let ( t , 0 ≤ t ≤ 1) denote the standard normalized Brownian excursion ([23, Chapter XII]). The CRT is the random isometry class T 2 , defining a random variable in (T , d GH ). Note the convention to use twice the distance d . This is only a matter of convenience, and some references call "CRT" the metric space T .

Main result

We are now ready to state our main theorem. As a motivation, we start with a well-known result. Let P n be the uniform distribution on T n . The following result is a re-interpretation in terms of R-trees of a result of Aldous, see [START_REF] Aldous | The continuum random tree[END_REF] and [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF].

Proposition 1 Let T n be a random variable with distribution P n . Then

1 √ 2n T n (d) -→ n→∞ T 2 ,
where the convergence in distribution holds with respect to the Gromov-Hausdorff topology on M .

Our main result is an analog of the previous statement for trees. We let P n be the uniform distribution on T n .

Theorem 1 Let T n be a random variable with distribution P n . Then

c √ 2n T n (d) -→ n→∞ T 2 ,
where the convergence in distribution holds with respect to the Gromov-Hausdorff topology on M , and where c is given in [START_REF] Bertoin | Lévy processes[END_REF].

In particular, one will notice that a tree with law P n is typically about c times smaller (indeed c is greater than 1) than a typical P n -distributed plane tree (in passing, a quick computation based on Figure 1 shows that the mean height of a random variable with distribution P 4 is 14/5, while it equals 5/2 for a random variable with distribution P 4 ).

Contents of the paper and strategy of the proof of Theorem 1

In Section 2.1 we present a trimming operation for plane and unordered trees: the idea is to keep in a large plane tree with n leaves, only the vertices having at least ⌊ǫn⌋ descendants, which in turn can be encoded as a plane tree with edge-lengths, called the ⌊ǫn⌋-skeleton. A similar operation is defined on (unordered) trees in Section 2.2.

This operation is the main tool in our study. Asymptotically, the ⌊ǫn⌋-skeletons of P n and P ndistributed random variables will turn out to be directly comparable thanks to two very similar "local limit theorems" (Propositions 2 and 3), which give the scaling limits for the densities of these ⌊ǫn⌋-skeletons. These are obtained via combinatorial arguments developed in Sections 2.1 and 2.2.

These local limit theorems form the cornerstone of our study, and they are, at the intuitive level, the main explanation of the similar asymptotic behaviour of both families of trees. However, these results are not sufficient by themselves to entail Theorem 1. First, because the pointwise convergence of the density of a random variable is not sufficient to get convergence in distribution. We need to check that the limiting formulas in Propositions 2 and 3 define indeed probability distributions, i.e. to ensure that no mass disappears, or goes into a singular part. A second problem is that the convergence of the ⌊ǫn⌋ skeleton does not imply immediately the convergence of the non-trimmed trees under P n : we have to rule out the possibility that the parts of the trees that have been removed in the trimming procedure are thin "hair" of very large diameter although they contain a small amount of leaves (at most ⌊ǫn⌋). The identification of the limit is then a consequence of Proposition 1.

Let us describe more precisely these steps. The local limit result, Proposition 2, concerning the ⌊ǫn⌋-skeleton under P n , is shown to imply a convergence in distribution in Section 4. To this end, we make a strong use of the convergence of the contour process of a P n -distributed plane tree toward the Brownian excursion (Proposition 4). This involves a careful translation of our trimming operations in terms of operations on contour processes, and more generally on excursion functions seen as encoding R-trees as explained in Section 1.4. This is done in Sections 4.1 and 4.2.

As a matter of fact, in the plane case, the convergence of contour processes under P n is sufficiently robust to entail that of the ⌊ǫn⌋-skeleton to an ǫ-trimmed version of the CRT, described in term of the Brownian excursion (Corollary 1 in Section 4.3, and Sections 4.5 and 4.6). This gives the wanted convergence in distribution for ⌊ǫn⌋-trimming of a P n -distributed plane tree, and a simple comparison argument (thanks to Propositions 2 and 3), allows to prove that ⌊ǫn⌋-skeletons under P n converges also in distribution toward the same limit, up to a constant factor (Section 5.1).

Finally, we show in Section 5.2 that the trimmed versions are not too far from the whole tree with high probability: this part, which amounts to controlling the maximal height of all the subtrees appearing in the difference between a tree and its ⌊ǫn⌋-skeleton relies on moment estimates for the height of a P n -distributed tree, obtained by Broutin & Flajolet [START_REF] Broutin | The height of random binary unlabelled trees (extended abstract)[END_REF].

Acknowledgments. We are grateful to Mathilde Weill, whose help and insights in the early stage of this work have been very important. Grateful thanks are due to Nicolas Broutin and Philippe Flajolet for the interest they have taken in the problem of estimating the total height of a random P n -distributed tree, and for keeping us informed of the progress of their work, an extended abstract of which can be found in [START_REF] Broutin | The height of random binary unlabelled trees (extended abstract)[END_REF]. These results are indeed crucial in obtaining a tightness argument of the kind of Lemma 22.

Trimming trees

Our main tool to prove Theorem 1 will be to use a mass-trimming of the tree from the leaves. The underlying idea is that the combinatorics of the resulting tree are a lot more tractable, and show quite easily the universal aspects of different tree models, at least for binary trees. In this perspective, we are going to compare the trimming of plane trees and trees with respective distributions P n , P n .

The case of plane trees 2.1.1 a-trimmed tree

Let Ø be a plane tree, and for a ≥ 0, the a-trimmed tree is defined to be Ø

[a] = {u ∈ Ø : |Ø v | > a ∀ v ≺ u} .
Of course, to check that u ∈ Ø[a] it suffices to verify that |Ø v | > a where v is the parent of u. However, with our definition, Ø[a] always contains ∅, and obviously defines a plane tree. For instance, Ø[0] = Ø [START_REF] Aldous | The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin's local time identity[END_REF] = Ø (see also Figure 5). From now on we assume that a ≥ 2.

u is such that u1, u2 are both in L(Ø[a]), then it must hold that |Ø u1 | + |Ø u2 | > a (otherwise |Ø u | ≤ a so u1 could not be in Ø[a]
). The following lemma is a (straightforward) kind of converse to this observation.

Lemma 1 Let Ø 0 ∈ T, and (Ø (u) , u ∈ L(Ø 0 )) ∈ T L(Ø 0
) be a family of plane trees indexed by the leaves of Ø 0 , such that

• |Ø (u) | ≤ a for every u ∈ Ø, • |Ø (u1) | + |Ø (u2) | > a whenever u1, u2 ∈ L(Ø), therefore Ø 0 = Ø[a], where Ø = Ø 0 ∪ u∈L(Ø 0 ) uØ (u) . (7)
Consequently, a plane tree can be recovered in a one-to-one fashion from its a-trimming Ø[a] and an appropriate family of plane trees indexed by the leaves of Ø[a]. Define

T ≤a = {Ø ∈ T : |Ø| ≤ a},
the set of trees having at most a leaves, and

F a = { = (Ø, Ø ′ ) : Ø, Ø ′ ∈ T, |Ø| ∨ |Ø ′ | ≤ a, |Ø| + |Ø ′ | > a} .
A pair of the form = (Ø, Ø ′ ) is called a forest with two tree components, namely With this notation, we can rewrite [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF] as

1 = Ø, 2 = Ø ′
Ø = Ø 0 ∪ u∈S(Ø 0 ) uØ (u) ∪ u∈I 0 (Ø 0 ) u (u) ,
where the Ø (u) are in T ≤a and the (u) are in F a . In the forthcoming Lemma 3, we will make one further adjustment by ordering the elements of S(Ø 0 ) and I 0 (Ø 0 ) in lexicographical order, allowing us to label the families Ø (u) and (u) by integers rather than elements of Ø.

In order to study the shape of Ø[a] we introduce two operations on plane trees. Their aim is to encode in some sense Ø[a]: when a is large, the tree Ø[a] will have few true branching points

(#I 2 (Ø[a]
) is small), and a lot of skeleton leaves. The operations defined below in the context of general binary plane trees will then be applied on Ø[a] in Section 3.

Contraction of a plane tree

We first define an operation of concatenation. Assume given two plane trees Ø, Ø ′ , together with labels

x = (x u , u ∈ Ø), x ′ = (x ′ u , u ∈ Ø ′
) taking their values in some set E (typically, E will be the set Z + ). For x ∈ E, we define the concatenation

Ξ(x, (Ø, x), (Ø ′ , x ′ )) = (Ø ′′ , x ′′ ) ,
to be the tree having its root ∅ marked by x, and having the two marked tree Ø and Ø ′ as subtrees rooted at the children 1 and 2 of ∅. Formally

Ø ′′ = {∅} ∪ 1Ø ∪ 2Ø ′ ∈ T , and x ′′ = (x ′′ u , u ∈ Ø ′′ ) is defined by x ′′ ∅ = x , x ′′ 1u = x u , u ∈ Ø , x ′′ 2u = x ′ u , u ∈ Ø ′ .
Let us introduce the contraction application Π. The action of Π amounts to suppressing all the leaves of Ø, and then to contracting the maximal chains of vertices with degree 2 in the resulting graph (while keeping its planar structure). This provides a new labeled binary tree Ø ′ , the label of a vertex being simply the size of the chain that has been contracted underneath it (see Figure 6).

Let us be formal: for every Ø ∈ T \ {{∅}} (so that |Ø| ≥ 2), we define the contraction of Ø as follows. Let U (Ø) be the vertex with minimal height having degree 2 in Ø satisfying

∀v ≺ u , |Ø v1 | = 1 xor |Ø v2 | = 1 ,
where xor denotes exclusive or. It must be understood that if no such v as in the definition exists, then U (Ø) = ∅. Then the contraction of Ø ∈ T is defined in a recursive way as a plane tree with integer labels Π(Ø) = (×, x) as follows:

• If |Ø U (Ø)1 | = |Ø U (t)2 | = 1 then × = {∅} and x ∅ = |U (Ø)| , • otherwise, it must hold that |Ø U (Ø)1 | ∧ |Ø U (Ø)2 | ≥ 2, and we set Π(Ø) = Ξ(|U (Ø)|, Π(Ø U (Ø)1 ), Π(Ø U (Ø)2 )) . (8)
This branching contraction procedure, together with the interpretation given at the beginning of Section 2.1.2, entails that the vertices of Π(Ø) are naturally associated with a subset of the vertices of Ø via an application π Ø . As one may observe on Figure 6, the branching structure of Π(Ø) is given by the relative positions of the vertices in I 2 (Ø) ∪ I 0 (Ø). More precisely, we let π Ø (∅) = U (Ø), and recursively, for u ∈ I(Π(Ø)), we set

π Ø (u1) = U (Ø πØ(u)1 ) , π Ø (u2) = U (Ø πØ(u)2 ) . ( 9
) 4 0 0 0 0 1 1 1 1 2 2 5
Figure 6: The plane tree Ø ′ = Ø [START_REF] Aldous | The continuum random tree. II. An overview[END_REF] (where Ø is given in Figure 5). The elements of I 2 (Ø ′ ) (repre- sented with squares), and elements of I 0 (Ø ′ ) (represented by small black discs) are the vertices giving the structure of the projection Π(Ø ′ ) = (×, x), represented on the third picture.

It is easy to check that π Ø is injective, with image I 2 (Ø) ∪ I 0 (Ø). More precisely, if × = Π(Ø), then

π Ø (I(×)) = I 2 (Ø) and π Ø (L(×)) = I 0 (Ø).
Moreover, π Ø is increasing for the lexicographical order. Noticing that the plane trees contracted to the tree ({∅}, x) are exactly those of the form {v, v1, v2 : v u} for a word u ∈ {1, 2} x , and using an induction argument, one may prove straightforwardly the following Lemma:

Lemma 2 Let × be a binary plane tree and x = (x u , u ∈ ×).

Then #Π -1 (×, x) = 2 |x| ,
where |x| = u∈× x u .

2.1.3 Encoding of the a-trimming of Ø: the a-skeleton of Ø

We now go one step further in our simplification of trees. Let (×, x) = Π(Ø[a]) be the contracted tree associated with the a-trimming of Ø. It is implicit that Π(Ø[a]) is indeed well-defined, meaning that |Ø[a]| ≥ 2, and this is equivalent to the fact that |Ø| > a, which is thus assumed.

For our purposes, it will be useful to assign to the leaves of × a second integer label, giving the total number of descendant leaves they had in the initial tree (corresponding to the size of the corresponding forest of pairs of trees). Formally, for u ∈ L(×),

y u = |Ø π Ø[a] (u) | ,
where the mapping π Ø[a] is as defined in [START_REF] Burago | A course in metric geometry[END_REF], and note that if

u ∈ L(×), it must hold that π Ø[a] (u) is an element of I 0 (Ø[a]
). Also, it holds that y v > a for every v ∈ L(×).

Definition 3

The triple (×, x, y) is called the a-skeleton of Ø, and is denoted by Sk a (Ø).

The general a-skeleton is thus a triple (×, x, y) such that

• × ∈ T, • x = (x u , u ∈ ×) is a vector of non-negative integers, • y = (y u , u ∈ L(×)) is a vector of integers in (a, 2a].
Summing up our study so far, we obtain the following result.

Lemma 3 Any (binary) plane tree Ø such that |Ø| > a can be obtained from the following data, in a one-to-one way:

1. an a-skeleton (×, x, y)

2. one of the 2 |x| elements Ø 0 ∈ Π -1 (×, x), 3. two sequences (Ø(i), 1 ≤ i ≤ |x|) ∈ T |x| ≤a and ( (i), 1 ≤ i ≤ |×|) ∈ F |×| a such that | (j)| = y v j , 1 ≤ j ≤ |×| , where v 1 ≤ . . . ≤ v |×| is the ordered list of leaves of |×|
by the formula:

Ø = Ø 0 ∪ |x| i=1 u i Ø(i) ∪ |×| j=1 l j (j) ,
where u 1 ≤ . . . ≤ u |x| and l 1 ≤ . . . ≤ l |×| respectively denote the ordered lists of elements of S(Ø 0 ) and I 0 (Ø 0 ), so in particular,

l i = π Ø 0 (v j ), 1 ≤ j ≤ |×|.

The case of trees

The procedure we adopt to trim a tree is similar to the one for plane trees. The reconstruction properties of a tree from its trimming are not as smooth as in the plane case though, as we will now see. First, we focus on what we call a labeled tree. A labeled plane tree is a pair (Ø, g) where Ø is a plane tree and g : Ø → E is a function from Ø to some set E. We define a new equivalence relation ≈ for labeled plane trees (Ø, g), by (Ø, g) ≈ (Ø ′ , g ′ ) if there exist permutations σ = (σ u , u ∈ Ø) as in Sect. 1.2 such that

Ø ′ = σ(Ø) , g ′ σ(u) = g u , u ∈ Ø .
A labeled tree is just an equivalence class for ≈. Such an equivalence class has at most 2 #I(t) = 2 |Ø|-1 representatives. When this upper-bound is attained, we say that the associated (rooted) labeled tree (or any of its plane representatives) has no symmetries. A simple criterion for having no symmetry is given by the following intuitive result. We say that a labeled plane tree (Ø, g) (or its ≈-equivalence class) is good if the components of g are pairwise distinct.

Lemma 4 A good labeled plane tree (Ø, g) has no symmetries.

Proof. Endow the label set E with any particular linear order. With any good labeled tree, one associates a particular planar representative (Ø, g) in which for every pair of vertices born from the same parent, the one with larger label is always the first child. The other planar representatives are obtained by deciding the order in which the two children of an interior vertex should be displayed, knowing that they can always be distinguished, as they have different labels, so there are exactly 2 |t|-1 of them.

Next, the reader will easily be convinced that Π and Ø → Ø[a] are class operations, meaning that if Ø ∼ Ø ′ then Ø[a] ∼ Ø ′ [a], and Π(Ø) ≈ Π(Ø ′ ). This validates the following Definition 4 We say that a plane tree Ø with |Ø| > a is a-good if it satisfies the following properties • the labeled tree Π(Ø[a]) is good, and

• for every u ∈ I 0 (Ø[a]), it holds that |Ø u1 | = |Ø u2 |.
We say that a tree t is a-good if one (hence any) of its planar representatives is.

If t is an a-good tree, we can choose in some non-ambiguous way a plane ≈-representative of its skeleton. To be specific, we will set Sk a (t) := (×, x, y) to be the representative such that x u1 > x u2 for every u ∈ I(×), and call it the a-skeleton of t. The general a-skeleton of an a-good tree (we naturally call such objects a-good skeletons) is thus a triple (×, x, y) such that • × is a plane tree,

• x = (x u , u ∈ ×) is such that (×, x
) is good and x u1 > x u2 whenever u ∈ I(×), and

• y = (y u , u ∈ L(×)) is such that y u ∈ (a, 2a] for u ∈ L(×).
The key observation is the following fact.

Lemma 5 Let Ø ∼ Ø ′ be a-good plane trees such that Ø[a] = Ø ′ [a], and such that for every

u ∈ I 0 (Ø[a]), one has |Ø u1 | > |Ø u2 | and |Ø ′ u1 | > |Ø ′ u2 | .
Then for every u ∈ L(Ø[a]), it holds that Ø u ∼ Ø ′ u .

Proof. Let σ = (σ u , u ∈ Ø) be such that σ(Ø) = Ø ′ . Assume that there exists u ∈ I(Ø[a]) such that σ u = 1 is not the identity. Let u be such a vertex of I(Ø[a]), with minimal height. Then

σ v = 1 for all v ≺ u, so since σ(Ø) = Ø ′ , it must hold that Ø u ∼ Ø ′ u . First assume u ∈ I 0 (Ø[a]), then by assumption, it holds that |Ø u1 | > |Ø u2 | and |Ø ′ u1 | > |Ø ′ u2 |.
Since Ø u ∼ Ø ′ u and σ u = 1, this entails that Ø u1 ∼ Ø ′ u2 (and Ø u2 ∼ Ø ′ u1 ), a contradiction with the order on the sizes of these trees.

A second possibility is that u ∈ I 1 (Ø), so that u1 or u2 is in S(Ø[a]). Assuming e.g. that the first case holds, this says that u2 is in

I(Ø[a]), but since σ u = 1 this says that u1 ∈ I(Ø ′ [a]) = I(Ø[a]) which is disjoint from S(Ø[a]), a contradiction.
The last possibility is that u ∈ I 2 (Ø[a]), and we

let v = π Ø[a] (u) ∈ I(×), where (×, x) = Π(Ø[a]
). But the fact that σ u = 1 has the effect of switching the values of x v1 , x v2 in the (good) labeled plane tree (×, x). Since these values are distinct, this contradicts Ø

[a] = Ø ′ [a].
This shows that σ u = 1 for every u ∈ I(Ø[a]) = I(Ø ′ [a]). The conclusion follows easily by noticing that Ø ′ u = σ (u) (Ø u ) for every u ∈ I(Ø[a]), where by definition, σ (u) = (σ uv : v ∈ I(Ø u )).

We now choose, once and for all, a section Π -1 0 of Π, i.e. a mapping from the set of a-skeletons to T such that Π • Π -1 0 is the identity.

13

Starting from an a-good tree t with a-skeleton (×, x, y), we can always choose a plane representative Ø of t such that Ø[a] = Π -1 0 (×, x). There might be many such choices of Ø, but by Lemma 5, we know there is a well-defined family of trees t (u) , u ∈ L(Ø 0 ), such that |t (u1) | > |t (u2) | whenever u ∈ I 0 (Ø 0 ), and such that t is the ∼-equivalence class of the plane tree (recall that s : T → T is a section)

Ø[a] ∪ u∈L(Ø 0 ) us(t (u) ) .
More precisely, letting

T ≤a = {t ∈ T : |t| ≤ a} and F a = {(t, t ′ ) : a ≥ |t| > |t ′ |, |f | := |t| + |t ′ | > a} , the trees t (u) , u ∈ S(Ø[a]
) lie in T ≤a , while the pairs (t (u1) , t (u2) ) for u ∈ I 0 (Ø[a]) lie in F a and satisfy

|t (u1) | + |t (u2) | = y π Ø[a] (u) . By convention, for f = (t, t ′ ) ∈ F a we let s(f ) = (s(t), s(t ′ )) ∈ F a .
Summing up our study, we get:

Lemma 6 Any a-good tree t can be obtained from the following data in a one-to-one way:

1. a good a-skeleton (×, x, y) such that x u1 > x u2 for every u ∈ I(×),

f (j) = (t(j), t ′ (j)) and |f (j)| = y v j , 1 ≤ j ≤ |×| ,
where v 1 , . . . , v |×| is the ordered list of elements of L(×), by letting t be the ∼-equivalence class of the planar tree

Ø = Ø 0 ∪ |x| i=1 u i s(t(i)) ∪ |×| j=1 l j s(f (j)) .
where Ø 0 = Π -1 0 (×, x), and u 1 , . . . , u |x| resp. l 1 , . . . , l |×| are the ordered lists of elements of S(Ø 0 ) and I 0 (Ø 0 ).

Local limit theorems for a-skeletons of trees

Let us draw the probabilistic consequences of our analysis of trimmed trees, starting with plane trees.

Plane trees

Let µ = (µ k ) k≥1 be the law of the size of a Galton-Watson tree with offspring distribution (δ 0 + δ 2 )/2; one has immediately

µ n = #T n 2 2n-1 , n ≥ 1 , (10)
and by Stirling's formula, the following equivalent holds:

µ n ∼ n→∞ 1 2 √ πn 3/2 . ( 11
)
We recall that conditioned on having n vertices, a Galton-Watson tree with offspring distribution (δ 0 + δ 2 )/2 is uniform on T n . Consider a probability space (Ω, F, P ) on which is defined an i.i.d. sequence X 1 , X 2 . . . of µ-distributed random variables.

Lemma 7 Let (×, x, y) be an a-skeleton. Then, seeing Sk a : Ø → Sk a (Ø) as a random variable,

P n (Sk a = (×, x, y)) (12) = 1 2 2|×|-1 µ n P   |x| i=1 X i = n -|y|, max 1≤i≤|x| X i ≤ a   u∈L(×) P (X 1 + X 2 = y u , X 1 ∨ X 2 ≤ a).
Proof. By Lemma 3, the number of plane trees admitting (×, x, y) for a-skeleton equals

2 |x| #    (Ø(1), . . . , Ø(|x|)) ∈ T |x| ≤a : |x| i=1 |Ø(i)| = n -|y|    u∈L(×) #{ ∈ F a : | | = y u } ,
which equals by ( 10)

2 |x|+2n-2|y|-|x| |×| j=1 2 2y i -2 × P   |x| i=1 X i = n -|y| , max 1≤i≤|x| X i ≤ a   u∈L(×) P (X 1 + X 2 = y u , X 1 ∨ X 2 ≤ a) ,
giving the result after dividing by #T n .

From this, we deduce an important limiting result (Proposition 2), which can be thought of a kind of "local limit theorem" for the ⌊ǫn⌋ skeleton of a P n -distributed element.

From now on, we fix a number ǫ ∈ (0, 1) and will usually omit its mention to allow lighter notations. The number ǫ will be allowed to move in the further Section 5.2 (also at the end of Section 4.6), but we will take care that there is no ambiguity at this moment.

For × ∈ T and M > 1/ǫ, set

I M (×) := (x, y) ∈ R × + × R L(×) + : x u ∈ (M -1 , M ) u ∈ × y u ∈ (ǫ + M -1 , 2ǫ) u ∈ L(×) . ( 13
)
We also define the family of functions a k (for k ≥ 0) on (0, ∞) by letting a 0 ≡ 1, a 1 (x) = (2

√ πx 3/2 ) -1 ½ x≥ǫ and for k ≥ 2, a k (x) = 1 (4π) k/2 u 1 ,...,u k-1 ≥ǫ ½ {u 1 +...+u k-1 ≤x-ǫ} du 1 . . . du k-1 (u 1 . . . u k-1 (x -u 1 -. . . -u k-1 )) 3/2 , x ≥ 0 , (14)
the latter being = 0 on [0, kǫ]. Let also

b(y) = 1 4π ǫ y-ǫ du (u(y -u)) 3/2 , y ∈ (ǫ, 2ǫ] .
Note that the functions a k , b depend tacitly on the parameter ǫ. Finally, we let

g(x) = exp(-1/(4x)) 2 √ πx 3/2 , x > 0 ( 15
)
be the density of the hitting time of 1/ √ 2 by a standard Brownian motion. This is the density of a stable law with parameter 1/2, which has Laplace transform

R + exp(-λx)g(x)dx = exp(- √ λ) , λ ≥ 0 .
It can also be expressed as the density function of the law of the total sum of the atoms of a Poisson random measure on R + , with intensity dx 2

√ πx 3/2 ½ {x>0} .
The following statement is a "local limit theorem" for a-skeletons of plane trees.

Proposition 2 Let × ∈ T. Then for every ǫ ∈ (0, 1) and M > 1/ǫ, the quantity sup

(x,y)∈Z × + ×Z L(×) + : (x/ √ n,y/n)∈I M (×) n 2|×|-1/2 P n (Sk ⌊ǫn⌋ = (×, x, y)) -ψ × (x/ √ n, y/n) ( 16
)
converges to 0 as n → ∞, where, setting z = 1 -|y|,

ψ × (x, y) = √ π 2 2|×|-2   ⌊z/ǫ⌋ k=0 (-|x|) k-2 k! z kǫ g z -u |x| 2 a k (u)du   v∈L(×) b(y v ) .
Although it is not apparent at first sight, the functions ψ × are equal to 0 whenever |×| > 1/ǫ. Indeed, in this case × cannot be the first component of the ⌊ǫn⌋-skeleton of a plane tree, since otherwise y n would have strictly more than 1/ǫ components of size at least ǫn, which is impossible.

By comparing this proposition with a statement like Proposition 1, it is tempting to interpret the functions ψ × , × ∈ T as density functions associated with a trimmed version of the CRT. We are going to make this more precise in Section 4.6.

In order to prove the proposition, write using Lemma 7,

n 2|×|-1/2 P n (Sk ⌊ǫn⌋ = (×, x, y)) = n -3/2 2 2|×|-1 µ n ×n P   |x| i=1 X i = n -|y|, max 1≤i≤|x| X i ≤ ⌊ǫn⌋   × u∈L(×) n 2 P (X 1 + X 2 = y u , X 1 ∨ X 2 ≤ ⌊ǫn⌋).
The asymptotics of the first term is √ π/2 2|×|-2 by [START_REF] Drmota | The shape of unlabeled rooted random trees[END_REF]. The two following Lemmas give an uniform approximation of the two other terms.

Lemma 8 For any M > 1/ǫ, we have the convergence

lim n→∞ sup l/n∈[ǫ+M -1 ,2ǫ] n 2 P (X 1 + X 2 = l, X 1 ∨ X 2 ≤ ⌊ǫn⌋) -b(l/n) = 0 , Proof. By definition, for l/n ∈ [ǫ + M -1 , 2ǫ] P (X 1 + X 2 = l, X 1 ∨ X 2 ≤ ⌊ǫn⌋) = ⌊ǫn⌋ j=l-⌊ǫn⌋ µ j µ l-j = 1 4π ⌊ǫn⌋ j=l-⌊ǫn⌋ 1 j 3/2 (l -j) 3/2 + o   ⌊ǫn⌋ j=⌊n/M ⌋ 1 j 3/2 (l -j) 3/2   ,
as n → ∞, where we have used the equivalent [START_REF] Drmota | The shape of unlabeled rooted random trees[END_REF], and the fact that l -⌊ǫn⌋ ≥ ⌊n/M ⌋. In turn,

1 4π ⌊ǫn⌋ j=l-⌊ǫn⌋ 1 j 3/2 (l -j) 3/2 = 1 4π ⌊ǫn⌋ l-⌊ǫn⌋ dx ⌊x⌋ 3/2 (l -⌊x⌋) 3/2 = 1 4πn 2 ⌊ǫn⌋/n (l-⌊ǫn⌋)/n dx (⌊nx⌋/n) 3/2 (l/n -⌊nx⌋/n) 3/2 .
This yields the result as the function (x, y) → x -3/2 (y -x) -3/2 is uniformly continuous on the compact set {(x, y) : (x, y -x) ∈ [(2M ) -1 , 2ǫ] 2 }, in which the terms (⌊nx⌋/n, l/n) of the above integral are constrained to lie (at least for n large enough).

Lemma 9 For M > 1, lim n→∞ sup l/ √ n∈[M -1 ,M ] m/n∈[0,1] nP l i=1 X i = m, max 1≤i≤l X i ≤ ⌊ǫn⌋ - k≥0 (-l/ √ n) k-2 k! m/n 0 du a k (u) g m/n -u (l/ √ n) 2 = 0 .
Proof. For events A, B j , j ∈ J with finite J, one has

E   ½ A j∈J (1 -½ B j )   = C⊆J (-1) #C E   ½ A j∈C ½ B j   ,
which is usually called the inclusion-exclusion principle. Using this and an elementary exchangeability argument, one has

nP l i=1 X i = m, max 1≤i≤l X i ≤ ⌊ǫn⌋ = n l k=0 (-1) k l k P l i=1 X i = m, min 1≤i≤k X i > ⌊ǫn⌋ ;
in order to let the reader follows more easily the step of the computations, we rewrite the RHS as

l k=0 (-1) k l k n k/2 × n k/2+1 P l i=1 X i = m, min 1≤i≤k X i > ⌊ǫn⌋ . (17)
Note that the sum can be reduced to its m/⌊ǫn⌋ first terms, because the constraint on the minimum of the first k variables X i forces k⌊ǫn⌋ < m. Thus, there are at most 1/ǫ non-zero terms in the sum, which is independent on n. It remains to show the (uniform) convergence of these individual terms. It is obvious that

l k /n k/2 ∼ (l/( √ n)) k /k! uniformly for l/ √ n ∈ [M -1 , M ].
To handle the probability term in [START_REF] Evans | Subtree prune and regraft: a reversible real tree-valued Markov process[END_REF], we introduce

A k (n, r) = P X 1 + . . . + X k = r, min 1≤i≤k X i > ⌊ǫn⌋ G(l, r) = P (X 1 + . . . + X l = r) ,
and write the quantity after the × sign, in [START_REF] Evans | Subtree prune and regraft: a reversible real tree-valued Markov process[END_REF],

m r=0 n k/2+1 A k (n, r)G(l -k, m -r) (18)
= m/n 0 n k/2+1 A k (n, ⌊rn⌋) √ n l 2 l 2 G(l -k, m -⌊rn⌋) dr .
In order to handle the terms involving G(l, m), we use the Gnedenko-Kolmogorov local limit theorem for sums of i.

In particular, in the expression [START_REF] Flajolet | Analytic Combinatorics[END_REF], we can omit the -k inside the G terms. On the other hand, we have lim

n→∞ sup 0≤m≤n n 1+k/2 A k (n, m) -a k (m/n) = 0 . (20)
Indeed, using [START_REF] Drmota | The shape of unlabeled rooted random trees[END_REF],

A k (n, m) ∼ j 1 ,...,j k >⌊ǫn⌋ j 1 +...+j k =m k l=1 1 2 √ πj 3/2 l = (4π) -k/2 dv 1 . . . dv k-1 ½ D(k,m) (v) m -k-1 i=1 ⌊v i ⌋ 3/2 k-1 i=1 ⌊v i ⌋ 3/2
, where by definition

D(k, m) = {(v 1 , . . . , v k-1 ) : min 1≤i≤k-1 ⌊v i ⌋∧(m-k-1 i=1 ⌊v i ⌋) > ⌊ǫn⌋}.
After the linear change of variables u i = v i /n, (20) boils down to a situation similar to Lemma 8, using the fact that the function

(u 1 , . . . , u k-1 , x) → (x -u 1 -. . . -u k-1 ) k-1 i=1 u i -3/2 is (uniformly) continuous on the compact (u 1 , . . . , u k-1 , x) : 0 ≤ x ≤ 1, min 1≤i≤k-1 u i ≥ ǫ/2, x -u 1 . . . -u k-1 ≥ ǫ/2 .
It is now easy to conclude, by ( 18), ( 19) and ( 20).

The case of trees

We now want to make a study parallel to the previous one in case of trees. We let

ν n = ̺ n #T n , n ≥ 1, ( 21
)
which defines a probability distribution on N thanks to (3), satisfying

ν n ∼ n→∞ c 2 √ πn 3/2 . ( 22
)
We now let (Ω, F, P ) be a probability space under which an i.i.d. sequence X ′ 1 , X ′ 2 , . . . of νdistributed random variables is defined. A (non-plane) tree analog of Lemma 7 states as follows. We let G a denote the set of a-good trees, and P a n := P( • ∩ G a ).

Lemma 10 Let (×, x, y) be an a-good skeleton. Then, seeing Sk a : t → Sk a (t) as a random variable,

P a n (Sk a = (×, x, y)) = 1 2 |×| ν n P   |x| i=1 X ′ i = n -|y| , max 1≤i≤|x| X ′ i ≤ a   × u∈L(×) P (X ′ 1 + X ′ 2 = y u , X ′ 1 ∨ X ′ 2 ≤ a, X ′ 1 = X ′ 2 ).
Proof. The proof goes like that of Lemma 7, applying Lemma 6 (and using ν instead of µ); the only important difference being that in the first displayed formula of the proof, we must replace plane trees with trees, the factor 2 |x| does not appear anymore, and we should consider pairs of trees (t, t ′ ) such that |t| + |t ′ | = y i and |t| < |t ′ |, the last constraint being absent in the plane case. This can be replaced by the constraint |t| = |t ′ | by introducing factors of 2, one for each 1 ≤ j ≤ |×|.

Besides the fact that the random variables X ′ i replace X i , the main difference with Lemma 7 is the presence of the event G a , and of the additional condition X ′ i = X ′ 2 in the last event. We can also make a comment on the fact that a prefactor of the form 2 -|×| replaces a factor 2 -2|×|+1 in Lemma 7: if we were concerned with the law of the non-plane version p(T n ) of a P n -distributed random variable T n , then for each (non-plane) possible skeleton (×, x, y) without symmetries, there would be exactly 2 |×|-1 plane skeletons contributing to its weight.

From this, one deduces a counterpart of Proposition 2 for (non-plane) trees. For × ∈ T, we let ĨM (×) be defined in a similar way as I M (×) in [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF], except that we further ask that the components of x are pairwise distinct, and are such that x u1 > x u2 whenever u ∈ I(×).

Proposition 3 Let × ∈ T. Then for every ǫ ∈ (0, 1) and M > 1/ǫ, the quantity

sup x,y∈Z × + ×Z L(×) + : (x/ √ n,y/n)∈ ĨM (×) n 2|×|-1/2 P ⌊ǫn⌋ n (Sk ⌊ǫn⌋ = (×, x, y)) -2 |×|-1 c 2|×|-1 ψ × (cx/ √ n, y/n) converges to 0 as n → ∞.
The proof follows exactly the same route as that of Proposition 2, using the two following intermediate lemmas.

Lemma 11 For any M > 1/ǫ, we have the convergence

lim n→∞ sup l/n∈[ǫ+M -1 ,2ǫ] n 2 P (X ′ 1 + X ′ 2 = l, X ′ 1 ∨ X ′ 2 ≤ ⌊ǫn⌋, X ′ 1 = X ′ 2 ) -c 2 b(l/n) = 0 , Lemma 12 For M > 1, lim n→∞ sup l/ √ n∈[M -1 ,M ] m/n∈[0,1] nP l i=1 X ′ i = m, max 1≤i≤l X ′ i ≤ ⌊ǫn⌋ - k≥0 (-cl/ √ n) k-2 k! m/n 0 du a k (u) g m/n -u (cl/ √ n) 2 = 0 .
In turn, the proofs of these lemmas are exactly the same as that of Lemmas 8 and 9. In the proof of Lemma 11, one just has to be careful of the extra c 2 term which comes in front of b, by using the equivalent [START_REF] Otter | The number of trees[END_REF] instead of [START_REF] Drmota | The shape of unlabeled rooted random trees[END_REF], and take into account the (asymptotically unimportant) fact that X ′ 1 , X ′ 2 are constrained to take distinct values. In the proof of Lemma 12, a similar remark holds for the function a k , which has to be replaced by c k a k , while the function g must be replaced by x → c -2 g(c -2 x), which is the density probability function of the total sum of masses of a Poisson process with intensity c dx 2

√ πx 3/2 ½ {x>0} .

Trimming continuous trees

We now want to have an interpretation of the functions ψ × in terms of the CRT. To this end, we first give another interpretation of the trimming operations discussed in Section 2.1.

Excursion functions and trimming

Recall that E is the set of continuous functions f defined on a compact interval [0, σ f ], that are non-negative, and satisfy f (0) = f (σ f ) = 0. With every f ∈ E, one associates an R-tree T f = [0, σ f ]/ ≡ f endowed with the distance d f , as in Section 1.4. This tree is naturally rooted (i.e. comes with a distinguished point) at the point ρ, which is the ≡ f -equivalence class of 0.

f I (t) = f (t + a) -f (a), 0 ≤ t ≤ σ f I = |I| .
The extremities a, b ∈ I encode the same point x = x I in T f , and f I encodes in turn an R-tree T f (x) := T f I , which is naturally interpreted as the subtree of T f rooted at x, i.e. the set of points y ∈ T f such that

d f (ρ, y) = d f (ρ, x) + d f (x, y) .
Yet otherwise said, this is the set of all descendants of x. We define the mass of T f (x) as

|T f (x)| = |I|.
Finally, for a ∈ (0, σ f ) a positive real number, we let

T [a] f = {x ∈ T f : |T f (x)| ≥ a} , which is also the closure of {x ∈ T f : |T f (x)| > a}, as is easily checked. The set T [a]
f is an R-tree which contains ρ, and is naturally rooted at this point.

As we will soon see, the operation

T f → T [a]
f is a continuous analog of the trimming operation

Ø → Ø[a] defined in Section 2.
First we present a slightly different way to look at T

[a]

f . For any f ∈ E and t ≥ 0, let O f (t) be the set of (non-empty) connected components of the open set {f > t}. It is convenient to imagine O f (t), t ≥ 0 as a fragmentation process, as the set {f > t} decreases from [0, σ f ] to ∅ when t moves from 0 to ∞. Note that, if I ∈ O f (t), then f I is well-defined and is positive on (0, σ f I ), and as above, I encodes a point x I in T f . This point will be in T

This identifies T

[a] f with the set of intervals I ∈ O f (t) with t ≥ 0 and |I| ≥ a. Note however that this identification is not one-to-one, as several intervals can encode the same point of T f .

From this, we can state the following Lemma allowing to control the Gromov-Hausdorff distance between T f and T

[a] f . Lemma 13 For every f ∈ E such that σ f > a, it holds that d GH (T f , T [a] f ) ≤ ω(f, a) , where ω(f, a) = sup |x-y|≤a |f (x) -f (y)| is the modulus of continuity of f , evaluated at a. In particular, T [a] f → T f for the Gromov-Hausdorff metric as a → 0. Proof. Since T [a] f is defined as a subset of T f , it suffices to show that any point of T f not in T [a] f
is at distance at most ω(f, a) from a point in T f . So let x be such a point. For every 0 < t < d f (ρ, x), there is a unique interval I x (t) ∈ O f (t) that contains some (hence all) representative of x in [0, σ f ] for the equivalence relation ≡ f . Since x is not in T

[a]
f is holds that |I x (t)| < a for t close enough to d f (ρ, x), so let t 0 be the infimum of all t's with this property. The intervals I x (t 0 -ǫ) decrease as ǫ → 0 to some interval I x (t 0 -) with length ≥ a, which encodes a vertex x 0 ∈ T

[a]

f . For every ǫ ∈ (0, d f (ρ, x) -t 0 ), the interval I x (t 0 + ǫ) has length < a, so easily

d f (x, x Ix(t 0 +ǫ) ) = f (s) -f (s ′ ) ≤ ω(f, a)
where s is a representative of x and s ′ is the left end of I x (t 0 + ǫ). Hence d f (x, x 0 ) ≤ ǫ + ω(f, a) for every ǫ > 0, giving the result.

The a-real skeleton description of T [a] f

Let us now derive an alternative representation of T

[a] f in terms of marked trees. To avoid trivialities we let

E a = {f ∈ E : σ f > a} ,
and will assume from now on that a > 0 and f ∈ E a so that in particular, T

[a]

f is non-empty and not reduced to a single point.

Assuming for a second that O f (0) = (0, σ f ), by continuity of f , O f (t) contains a single interval of length > a for every small enough t > 0. When t increases, this property will break down either because a second interval of length > a is created, or if no interval of length > a remains (see Figure 7). For every f ∈ E a , we let

t a (f ) = inf t ≥ 0 : sup I∈O f (t) |I| ≤ a ∈ [0, sup f ] ,
and By definition, for every t < s a (f )∧t a (f ), O f (t) contains a single interval I f (t) with |I f (t)| > a, and these decrease to a closed interval

s a (f ) = inf{t ≥ 0 : ∃I, I ′ ∈ O f (t) : I = I ′ , |I| > a and |I ′ | > a} ∈ [0, ∞] . a a a s a (f ) t a (f ) t a (f ) t
I f (s a (f ) ∧ t a (f )-) as t ↑ s a (f ) ∧ t a (f ). By convention, we let I f (0-) = (0, σ f ) in the case s a (f ) ∧ t a (f ) = 0. It is immediate to see that if s a (f ) < ∞ then s a (f ) < t a (f ). In the case s a (f ) = ∞, we let Y a (f ) := |I f (t a (f )-)| ≥ a .
In all cases, thanks to the description of

T [a] f in terms of the intervals of O f (t), one checks easily that the mapping t ∈ [0, s a (f ) ∧ t a (f )) → x I f (t) ∈ T f is an isometry, and that its image is exactly the set x ∈ T [a] f : d f (x, ρ) < s a (f ) ∧ t a (f ) .
This isometry can be extended continuously to the whole segment [0, s a (f ) ∧ t a (f )] by mapping its right end to x I f (sa(f )∧ta(f )-) . Otherwise said, for f ∈ E a , the "bottom" of the tree T

[a] f is made of a line segment (possibly reduced to the single point {ρ}). Moreover, if s a (f ) = ∞, then the above isometry admits all of T

[a] f as its image, so the latter is isometric to the segment [0, t a (f )]. So let us study what happens above level s a (f ) in the case s a (f ) < ∞.

We will be interested in a particular type of functions of E a . We say that x ∈ (0, σ f ) is a local minimum of f if f (x) ≤ f (y) for every y in an open neighborhood of x. Note that 0, σ f are purposely excluded with this definition. We let m f be the set of local minima of f .

For a > 0, we let E a be the set of functions f ∈ E such that for all x, y ∈ m f , x ≤ y such that f (x) = f (y), then x = y or min{f (t) : t ∈ [x, y]} < f (x). This condition implies that T f has only binary branching points. We also let E * a be the subset of E a made of those functions f such that a / ∈ closure({|I| : ∃t ≥ 0,

I ∈ O f (t)}) .
The sets E a , E * a are not nice from a topological perspective (in particular they are neither open nor closed), but it turns out that the mapping f → O f behaves well when restricted to these sets, as the following lemmas will show.

Lemma 14 Let a > 0 and f ∈ E a . 1. If s a (f ) = ∞ and f ∈ E * a , then Y a (f ) > a. 2. If s a (f ) < ∞, then O f (s a (f )) has exactly two components I (l) f , I (r) f with |I (l) f | ∧ |I (r) f | > a,
and these can be ordered so that sup

I (l) f = inf I (r) f . We then set f (l) = f I (l) f , f (r) = f I (r) f . Proof. 1. Assume s a (f ) = ∞. The possibility that Y a (f ) = a is excluded by definition of Y a (f )
and the fact that a is not a limit point of the set of lengths of intervals in ∪ t≥0 O f (t). Hence Y a (f ) > a.

2. Assume s a (f ) < ∞. Let I 0 = I f (s a (f )-). Necessarily, it must hold that f I 0 attains the value 0 at some point x of (0, σ f I 0 ), which is unique as it must correspond to a local minimum of f ∈ E a . Hence, O f I 0 (0) is made of two intervals I 1 , I 2 such that sup I 1 = inf I 2 = x, and inf I 1 = 0, sup I 2 = |I 0 |. These two intervals correspond to two intervals

I (l) f = (inf I 0 , x + inf I 0 ) , I (r)
f = (x + inf I 0 , sup I 0 ) , belonging to O f (s a (f )). By definition, the latter set contains two intervals with lengths > a, but

I (l) f , I (r)
f are the only possible ones, since an interval I ∈ O f (s a (f )) is a subinterval of some interval in O f (t) for t < s a (f ), so in order that |I| > a we must have I ⊂ I f (t) for every t < s a (f ), giving I ⊂ I 0 . Point 2. in Lemma 14 entails that in the case s a (f ) < ∞, the subtree of T f above the point x I f (sa(f )-) splits into two branches with masses > a. More precisely, if h > 0 and T , T ′ are two rooted R-trees with roots ρ, ρ ′ , we let Θ(h, T , T ′ ) be the R-tree obtained by identifying the roots ρ, ρ ′ and grafting to this point a line segment of length h. The latter can be easily formalized as the quotient of the disjoint union [0, h] ⊔ T ⊔ T ′ in which the point h ∈ [0, h] is identified with ρ and ρ ′ , and endowed with the proper distance function that extends the metrics on each of these components. The tree Θ(h, T , T ′ ) is naturally rooted at the point 0 ∈ [0, h]. Then if f ∈ E a and s a (f ), with the notations of Lemma 14,

T [a] f isom = Θ(s a (f ), T [a] f (l) , T [a] f (r) ) , (23)
while T

[a]

f is isometric to [0, t a (f )] if s a (f ) = ∞. Put together, these two facts allow a recursive construction of T f , which ends in a finite number of steps, since by definition, we have

σ f (l) ∨ σ f (r) ≤ σ f -a whenever s a (f ) < ∞.
This description can be further simplified as follows. Recall the definition of the concatenation operation Ξ for labeled plane trees, defined in Section 2.1. We can associate, in a recursive manner, a marked plane tree (×, g) = ζ(f, a) with g = (x, y) ∈ R × × R × with every f ∈ E a in the following way.

• If s a (f ) = ∞ then × = {∅}, x ∅ = t f (a), y ∅ = Y a (f ). • If s a (f ) < ∞ then ζ(f, a) = Ξ((s a (f ), 0), ζ(f (l) , a), ζ(f (r) , a)).
Note that every y-mark of an internal vertex is 0, so we simply forget these marks, identifying ζ(f, a) with the tree × with marks (x, [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]. With every pair of the form (×, x), where × ∈ T and x = (x u , u ∈ ×) ∈ (0, ∞) × , we associate an R-tree θ(×, x) in a natural way, by grafting segments with lengths x u according to the genealogy of ×.

y) ∈ R × × R L(×) . a s ∅ s ∅ s 1 s 1 t 0 t 0 t 1 t 1 t 2 t 2 Y 1 Y 1 Y 2 Y 2 Y 3 Y 3
Formally, we let θ((∅, x ∅ )) be the isometry class of [0, x ∅ ], and recursively, if

× = {∅}, θ(×, x) = Θ(x ∅ , θ((×, x) 1 ), θ((×, x) 2 )) ,
where by definition (×, x) u is the plane subtree × u with labels x u = (x uv , v ∈ × u ).

Lemma 15 Let f ∈ E a . Then T [a]
f and θ(ζ(f, a)) are isometric, where in the latter notation we do not take the y-marks into account.

To prove this, observe that θ(ζ(f, a)) is isometric to [0, t a (f )] if s a = ∞, and that

θ(ζ(f, a)) = Θ(s a (f ), θ(ζ(f (l) , a)), θ(ζ(f (r) , a))) ,
if s a (f ) < ∞, and compare with (23).

Continuity properties of the a-real skeleton

Lemma 16 Let a > 0 and (a n , n ≥ 1) be a positive sequence converging to a. Let (f n , n ≥ 1) be a sequence such that f n ∈ E an for every n, uniformly converging to f ∈ E * a . Then f . Let also ǫ ∈ (0, ((y -x)∧(z -y))/2). From the uniform convergence of f n to f , the minimum of f n over (x + ǫ, z -ǫ) is attained at some y , which converges to y as

1. If s a (f ) = ∞, then t an (f n ) → t a (f ), and Y an (f n ) → Y a (f ) as n → ∞. 2. If s a (f ) < ∞, then s an (f n ) → s a (f ) and (f (l) n , f (r) n ) → (f (l) , f (r) ) as n → ∞.
n → ∞ since f (v) > f (y) for all v ∈ [x + ǫ, y) ∪ (y, z -ǫ]. Let x n = sup{v < y n : f n (v) = f n (y n )} and z n = inf{v > y n : f n (v) = f n (y n )}. Then we have x n → x, z n → z as n → ∞.
Consequently, the intervals (x n , y n ), (y n , z n ) have both lengths > a n for n large enough, since y -x, z -y are > a and a n → a. It to note that for large n, there can be no two intervals of length > a n in O fn (t) for t < f n (y n ), in which case it will follow that s an (f n ) = f n (t n ) → f (t) = s a (f ), and the remaining properties are easy. But assuming that two such intervals exist, one of them must contain the interval (x n , z n ). By extracting subsequences, we may assume that the other interval is of the form

(x ′ n , z ′ n ) (with z ′ n -x ′ n > a n ) and that its extremities converge to (x ′ , z ′ ). By construction f (x ′ ) = f (z ′ ) ≤ s a (f ) and by definition, it holds that a = lim a n ≤ z ′ -x ′ ≤ a. Hence, (x ′ , z ′ ) is an interval of length a in O f (f (x ′ )) contradicting the fact that f ∈ E * a .
Corollary 1 Let a > 0 and (a n , n ≥ 1) be a positive sequence converging to a. Let (f n , n ≥ 1) be a sequence such that f n ∈ E an for every n, converging to

f ∈ E * a . Then ζ(f n , a n ) → ζ(f, a). Consequently, it holds that d GH (T [an] fn , T [a] f ) → 0.
The first part of this statement is obtained directly by combining Lemma 16 with the recursive definition of ζ(f, a). The last sentence follows from Lemma 15 and the easy fact that x → θ(×, x) is continuous from (0, ∞) × to (T , d GH ), for every × ∈ T. We now reconnect the previous discussion to our study of (discrete) trees. Let Ø ∈ T n . The contour exploration of Ø is the sequence (u Ø (0), . . . , u Ø (4n -4)) where u Ø (0) = ∅, and recursively, u Ø (i + 1) is the first child (in lexicographical order) of u Ø (i) not in {u Ø (0), . . . , u Ø (i)}, if such a child exists, or the parent of u Ø (i) otherwise. Each vertex is visited as many times as its degree in this sequence, which explains that the procedure stops at the (4n -4)-th step (indeed, each edge of the tree is visited twice, and there are 2n -2 edges in a binary tree with n leaves). A picture of this is to imagine a particle wrapping around the tree and moving at speed 1, starting from ∅ (see Figure 9).

The case of discrete trees

Definition 5

The contour process of Ø ∈ T n is the continuous function C Ø (t), 0 ≤ t ≤ 4n -4 whose graph is obtained by a linear interpolation between the points (i, |u Ø (i)|), 0 ≤ i ≤ 4n -4.

The contour process of Ø is also called the Dyck path encoding of Ø, as the latter can be recovered from its contour process. We will only be interested in recovering the metric space

(Ø, d Ø ) from C Ø . If v, w ∈ Ø, let 0 ≤ i, j ≤ 4n -4 be such that u Ø (i) = v, u Ø (j) = w. Then it is easy to see that |v ∧ w| = min i∧j≤k≤i∨j C Ø (k), independently of the choice of i, j. Therefore, d Ø (v, w) = C Ø (i) + C Ø (j) -2 min i∧j≤k≤i∨j C Ø (k) = d CØ (i, j) .
Of course, for integers i and j, d CØ (i, j) = 0 if and only if u Ø (i) = u Ø (j), which is equivalent to i ≡ CØ j, and so, the quotient space {0, 1, . . . , 4n -4}/ ≡ CØ endowed with the distance d CØ , is isometric to (Ø, d Ø ). Therefore Ø is isometric to a subset of T CØ , and it is immediate to check Figure 10 illustrates precisely the Lemma and the differences between the two points of view: the labels x of the vertices are the same up to 1 due to discrete-continuous artifacts, and a subtree of size k becomes a subpath of length 4(k -1); the proof below formalises these observations. Proof. We first note that since the plane trees that we consider are binary, it is immediate that the contour function C Ø is an element of E a for a < 4|Ø| -4, so that ζ(C Ø , a) is well-defined.

We will use the following alternative description of Sk a (Ø) = (×, x, y). Recall the definition of

π Ø[a] : × → I 2 (Ø[a]
). Then for u ∈ × and i ∈ {1, 2} such that ui ∈ ×, we have

x ui = |π Ø[a] (ui)| -|π Ø[a] (u)| -1 , counting the number of elements of I 1 (Ø[a]) comprised between π Ø[a] (u) and π Ø[a] (ui), while x ∅ = |π Ø[a] (∅)|.
Let us compare this with the construction of ζ(C Ø , 4(a -1)). For t ≥ 0 an integer, an interval I of O CØ (t) encodes a vertex u I of Ø, seen as a subset of T CØ as in the construction above. If the length of the interval is ≥ 4a -4, then the subtree Ø u I has at least a leaves, and consequently, this vertex belongs to Ø[a]. On the other hand, if the length is ≤ 4a -4, then none of the strict descendants of u I can be in Ø[a]. From these observations, it follows that in the case

s 4(a-1) (C Ø ) = ∞, it holds that Sk a (Ø) = ({∅}, s 4(a-1) (C Ø ), Y 4(a-1) (C Ø )/(4(a -1))) . In the case s 4(a-1) (C Ø ) < ∞, the vertex of Ø encoded by the interval I CØ (s 4(a-1) (C Ø )-) is the vertex π Ø[a] (∅) of Section 2.1.
2, and it holds that Sk a (Ø) = (×, x, y) satisfies x ∅ = s a (C Ø ). Now, it must be noted that the functions C

Ø , C

Ø defined as in Lemma 14 are not the contour functions of a binary plane tree, because they are strictly positive on their respective intervals of definitions. We let Ø (l) , Ø (r) be the binary plane trees that admit as contour functions the paths

C (l) Ø , C (r)
Ø , to which the first and last steps have been removed. Then Sk a (Ø) = Ξ(s 4(a-1) (C Ø ), Sk a (Ø (l) ), Sk a (Ø (r) )) .

Using this and an inductive argument, this indeed shows that

x u = x ′ u -1 , u = ∅ ,
with the notations of the statement, this -1 being due to the fact that the quantities s a , t a associated with the functions

C (l) Ø , C (r)
Ø , must be subtracted 1 in order to yield the number of skeleton leaves comprised between two elements of I 2 (Ø[a]).

In view of deriving scaling limits, define the rescaled contour function of Ø by

c Ø (s) = C Ø (4(|Ø| -1)s) 2|Ø| , 0 ≤ s ≤ 1 ,
and the n-rescaled skeleton as

Sk (n) a (Ø) = (×, n -1/2 x, n -1 y) ,
where (×, x, y) = Sk a (Ø). Then by Lemma 17, it holds that for Ø ∈ T n ,

ζ c Ø , 4(a -1) 4(n -1) = (×, x ′ , n -1 y),
where for any u ∈ ×, dxdy ψ × (x, y) .

|n -1/2 x u -x ′ u | ≤ n -1/2 .
Letting M → ∞, the sets I M (×) respectively increase to {(x, y) ∈ (0, ∞) × × (ǫ, 2ǫ) L(×) }, and we know from Lemma 18 that the law of ζ(2 3/2 , ǫ) is supported on the union of such sets as × ranges in T. Therefore, the probability on the left-hand side converges to 1 as M → ∞, which yields the first statement. The fact that Ψ is the law of ζ(2 3/2 , ǫ) is a simple adaptation of the previous argument, and is omitted.

Proof of Theorem 1

We now finally embark in the proof of our main results. This will be done in two steps: first, we prove that the ⌊ǫn⌋-trimmed tree obtained from a P n -distributed tree converges to the ǫtrimming of the (appropriately scaled) CRT. Then we show that the trimmed tree is "not too far" from the original tree.

Convergence of the a-skeleton under P n

The main result of this section is 29 Now, using this fact after conditioning on the sizes r i , i ≥ 1,

P n d GH (t, t[⌊ǫn⌋]) > η √ n ≤ P n (∃ i ≥ 0 : H(t i ) ≥ η √ n) ≤ E n i P r i (H ≥ η √ n) ≤ Cη -2 E n i (r i /n) 2
≤ Cη -2 ǫ .

In the last step, we wrote (r i /n) 2 ≤ ǫ(r i /n) and summed over i. Hence the result.

To conclude the proof of Theorem 1, simply apply the following and last lemma, with X n = c(2n) -1/2 t under P n , X (ǫ) n = c(2n) -1/2 t[⌊ǫn⌋] under P n , X (ǫ) = θ(ζ(2 , ǫ)) and X = T 2 . The hypotheses are enforced, by Lemma 22, and since X (ǫ) n → X (ǫ) in distribution (Proposition 5) and X (ǫ) → X a.s. as a consequence of Lemma 13.

Lemma 23 On some probability space (Ω, F, P ), let X n , X, X (ǫ) n , X (ǫ) , n ≥ 1, ǫ ∈ (0, 1) be random variables with values in some metric space (Z, d) such that X (ǫ) n → X (ǫ) in distribution as n → ∞, X (ǫ) → X in distribution as ǫ → 0, and

lim ǫ→0 sup n≥1 P (d(X n , X (ǫ) n ) > η) = 0 .
Then X n → X in distribution as n → ∞.

Proof. Let F be closed in Z, then

P (X n ∈ F ) ≤ P (X n ∈ F, d(X n , X ǫ n ) ≤ η) + δ (ǫ) η ,
where δ (ǫ) η = sup n P (d(X n , X

n ) > η). Thus P (X n ∈ F ) ≤ P (X (ǫ)

n ∈ F η ) + δ (ǫ)
η , where F η = {z ∈ Z : d(z, F ) ≤ η} is closed. Since X (ǫ) n → X (ǫ) in distribution as n → ∞, it holds that lim sup n P (X n ∈ F ) ≤ P (X (ǫ) ∈ F η ) + δ (ǫ) η , and taking the limsup as ǫ → 0, appealing to the convergence in distribution X (ǫ) → X, and finally, letting η → 0, this gives lim sup n P (X n ∈ F ) ≤ P (X ∈ F ) as wanted.

Final comments

To our knowledge, the present work is the first to deal with the full scaling limit picture of a random non-labeled, non-plane tree. As was predictable from the known enumeration results, and conjectured by Aldous, the CRT is the object that arises as the limit of uniform random binary rooted trees.

In this paper, we have purposely restricted our attention to binary trees in order to make the combinatorial arguments as simple as possible when considering trimming of trees. However, it is to be expected that the kind of methods used to prove statements like Proposition 5 will still work for more general families of unordered trees, for instance trees which are 'at most m-ary' for some 2 ≤ m ≤ ∞, as considered in Otter's work [START_REF] Otter | The number of trees[END_REF] (the case m = ∞ is sometimes referred to as Pólya trees). What is to be expected is that after a trimming by a mass ⌊ǫn⌋ a uniform tree with n vertices, the tree that one obtains will be binary with a probability close to 1, because the CRT is itself binary. In view of the probabilistic approach taken in Section 3.2, it seems also possible to adapt this method to more general families of trees, i.e. of trees satisfying an enumeration scheme similar to (4).

We have not gone through the details of this approach in full generality, though, for several reasons. One of these is that it needs a result similar to that of Broutin and Flajolet [START_REF] Broutin | The height of random binary unlabelled trees (extended abstract)[END_REF] for more general trees. This might appear as a minor issue, as combinatorial methods are probably robust enough to be generalizable to more general situations. For instance, while completing the present work, we became aware of the paper [START_REF] Drmota | The shape of unlabeled rooted random trees[END_REF] by Drmota and Gittenberger, in which moments estimates of the kind of [START_REF] Broutin | The height of random binary unlabelled trees (extended abstract)[END_REF] are obtained for the case of Pólya trees. However, one should note that the results of [START_REF] Broutin | The height of random binary unlabelled trees (extended abstract)[END_REF] give much more than what is actually needed to prove Lemma 22, i.e. precise moment estimates, or the derivation of the exact scaling limit of the height. From the probabilistic viewpoint, it would be very interesting to be able to decide in a slicker way whether this 'tightness' property of Lemma 22 holds in a general setting. Note in particular that a derivation of the scaling limit for the total height of the tree follows from Theorem 1:

P n (c H ≥ x √ 2n) -→ n→∞ P (2 max ≥ x) = 2 k≥1
(k 2 x 2 -1)e -k 2 x 2 /2 , for x > 0 (for the last inequality, see e.g. [START_REF] Durrett | Functionals of Brownian meander and Brownian excursion[END_REF]), which is [8, Theorem 1].

Figure 1 :

 1 Figure 1: Representation of the 5 rooted binary plane trees with 4 leaves. There are 2 (rooted) binary trees with 4 leaves (the 4 last ones on the picture representing the same unordered tree).

Figure 2 :

 2 Figure 2: A plane tree Ø such that L(Ø) = {111, 112, 12, 21, 22}, I 2 (Ø) = {∅}, I 0 (Ø) = {11, 2}, I 1 (Ø) = {1}, S(Ø) = {12} .

Figure 5 :

 5 Figure 5: A plane tree Ø; the tree Ø[a] with a = 3, i.e. those vertices whose fathers have 4 leaves at least among their descendants, is represented by the black vertices

  with natural notations. The size | | is then defined as | | = |Ø| + |Ø ′ |, and by convention, for u ∈ U and = (Ø, Ø ′ ), we let u be the set {u} ∪ u1Ø ∪ u2Ø ′ .

2 .

 2 two sequences (t(1), . . . , t(|x|)) ∈ T |x| ≤a and (f (1), . . . , f (|×|)) ∈ F |×| a such that

l 2 G

 2 i.d. random variables in the domain of attraction of a stable (1/2) distribution [7, Theorem 8.4.1], giving that lim l→∞ sup m≥0 (l, m) -g(m/l 2 ) = 0 .

20 For

 20 I a real interval, we let |I| = sup Iinf I denote its length. For every (nonempty, open or closed) interval I, a = inf I and b = sup I, such that inf I f = f (a) = f (b), we can define a new excursion function f I ∈ E by

f

  if and only if |I| ≥ a.

Figure 7 :

 7 Figure 7: The set of intervals O f (t) (left), the case s a (f ) < +∞ (middle), the case s a (f ) = +∞ (right).

Figure 8 :

 8 Figure 8: Representation of ζ(f, a)

Proof. 1 . 2 .

 12 This comes immediately from the uniform convergence of f n → f , and does not use the fact that f ∈ E * a . Let x = inf I

Figure 9 :

 9 Figure 9: A plane tree and its contour process.

Figure 10 :

 10 Figure 10: Comparison between the encoding via the skeleton and via the function ζ. On the first line the trimmed tree Ø[3] and the skeleton representation. On the second line, representation of ζ(C Ø , 12). The trimming operation discussed in Section 2.1 has a direct interpretation in terms of trees of the form ζ(f, a), when f is the contour function of a plane tree. Lemma 17 Let a > 0 be an integer, and Ø ∈ T n . If Sk a (Ø) = (×, x, y) then ζ (C Ø , 4(a -1)) = (×, x ′ , y ′ ) ,

For

  every ǫ ∈ (0, 1), we can define a probability measure Ψ(d(×, x, y)) on {(×, x, y) :× ∈ T, x ∈ R × , y ∈ R L(×) } by the formula Ψ, ψ × (x, y)F (×, x, y) ,for every non-negative measurable function F . Then Ψ is the law of ζ(2 3/2 , ǫ).Proof. Fix × ∈ T. Recall the definition (13) of the open set I M (×) ⊂ R × + × R L(×)+ . Then, defining ⌊x⌋ by taking integer parts componentwise, we haveP n Sk (n) ⌊ǫn⌋ ∈ ×∈T {(×, x, y) : (x, y) ∈ I M (×)} = ×∈T I M (×) dxdy n 2|×|-1/2 P n (Sk ⌊ǫn⌋ = (×, ⌊x √ n⌋, ⌊ny⌋)) .The lim inf of the left-hand side as n → ∞ is at leastP (ζ(2 3/2 , ǫ) ∈ ×∈T {(×, x, y) : (x, y) ∈ I M (×)),because of Corollary 2 and well-known properties of weak convergence of probability measures. And the right-hand side converges to ×∈T I M (×) dxdy ψ × (x, y) , because of the uniform convergence stated in Proposition 2. This yields P ζ(2 3/2 , ǫ) ∈ ×∈T {(×, x, y) : (x, y) ∈ I M (×)} ≤ ×∈T I M (×)

There is a slight difference in the article of Otter, as he considers trees with vertex-degrees either equal to 1, 2 or

It is easy to see that this is actually equivalent to the family of binary unordered trees, by adding "ghost" edges at every vertex with only one child.

Scaling limits

We now discuss the scaling limits of the trimmed tree. The core result is the counterpart of Proposition 1 (in fact, this is a stronger result) in terms of contour processes; it is due to Aldous [START_REF] Aldous | The continuum random tree[END_REF] (see also [START_REF] Marckert | The depth first processes of Galton-Watson trees converge to the same Brownian excursion[END_REF]) Proposition 4 Let T n , n ≥ 1 be random variables with respective distributions P n . Then c Tn converges to 2 , the normalized Brownian excursion, in distribution for the topology of uniform convergence on C([0, 1]).

Corollary 2 For every ǫ ∈ (0, 1), it holds that Sk Note the normalization in front of the Brownian excursion in the last statement, which comes from the fact that the x-components of the rescaled skeleton are obtained by dividing by √ n the components of the skeleton (instead of √ 2n). From Corollary 1 and the discussion after Lemma 17, this is an immediate consequence of the following Lemma 18 For every ǫ ∈ (0, 1), it holds that with probability 1, is an element of E * ǫ .

Proof. The fact that the local minima are realized once (a.s.) is a well-known property of the Brownian excursion, which is inherited from the analogous property for Brownian motion, and the property of positivity of over (0, 1) comes from the definition. Hence a.s. belongs to E ǫ . It remains to show that ǫ is not a limit point of the lengths of intervals of ∪ t≥0 O (t). To show this, we use the fact from [START_REF] Bertoin | Self-similar fragmentations[END_REF] that if U is a uniform random variable in [0, 1], independent of , then for 0 ≤ t < U , the unique element I U (t) of O (t) containing U is such that (|I U (t)|, t ≥ 0) has same distribution as (exp(-ξ τ (t) ), t ≥ 0), where

• ξ is a subordinator [START_REF] Bertoin | Lévy processes[END_REF], with Lévy measure e x dx/(e x -1) 3/2 , and • τ is the inverse of the increasing, continuous process t → t 0 exp(-ξ s /2)ds.

In particular, the closure C of the range of (|I U (t)|, t ≥ 0) is the image of the closure of the range of ξ by x → e -x , to which has been adjoined the point 0. By well-known properties of subordinators [START_REF] Bertoin | Lévy processes[END_REF], points are polar for ξ, so that ǫ is not in C, almost-surely. Since U is uniform and independent of , the wanted property easily follows -for instance, one can note that a.s.

and independent of , and with the obvious notation for I U i (t).

Interpretation in terms of ψ ×

Now we reconnect the previous discussion to the functions ψ × defined in Section 3.

Lemma 19

For every ǫ ∈ (0, 1), the functions ψ × , × ∈ T form a "probability density", in the sense that

Proposition 5 For every ǫ ∈ (0, 1), the random variables

, where is the standard Brownian excursion.

To prove this, we study the scaling limit of t → Sk ⌊ǫn⌋ (t) under P n . First, we define a probability measure, by

where

The fact that Ψ • , 1 = 1 is a consequence of the fact that Ψ • charges only good labeled trees (×, x, y), and Lemma 4, because of the presence of the factor 2 |×|-1 . The following is the analog of Corollary 2 (and the end of Lemma 19) in the context of (non-plane) trees.

Lemma 20 For every ǫ ∈ (0, 1), the law of t → Sk Proof. For × ∈ T and x ∈ (0, ∞) × , y ∈ (0, ∞) L(×) , we set

Using Proposition 3, we have that g n (×, x, y) → 2 |×|-1 c 2|×|-1 ψ × (cx, y) pointwise, and uniformly over × ĨM (×) for every M > 0. Consequently, for every M > 0, it holds that

dxdyg n (×, x, y)

Therefore, lim P n (G ⌊ǫn⌋ ) = 1, and by the so-called Scheffé Lemma, it holds that g n converges in L 1 . Letting F be a uniformly continuous function bounded by K, it holds that

converges to Ψ • , F , using the uniform continuity of F and the L 1 -convergence of g n .

Lemma 21 If (×, x, y) and (× ′ , x ′ , y ′ ) are respectively Ψ and Ψ • distributed, then θ(×, x)/c and θ(× ′ , x ′ ) have the same distribution.

Proof. Note that θ(×, x) = θ(× ′ , x ′ ) whenever (×, x) ≈ (× ′ , x ′ ), i.e. whenever these two plane labeled trees represent the same labeled tree. Therefore, for a given a-good tree t, θ(Sk a (t)) = θ(× ′ , x ′ ) for any of the 2 |×|-1 choices of (× ′ , x ′ ) ≈ Sk a (t), and there are exactly 2 |×|-1 of them. The conclusion follows easily from the definition of Ψ • and the fact that θ is a continuous function.

Proof

Tightness

By Proposition 5, we know that the ⌊ǫn⌋-trimming of a P n -distributed tree is close in distribution, when n is large, to θ(ζ(2 , ǫ)). In turn, the latter is close to the CRT T = T 2 when ǫ is small, by Lemma 13. On an intuitive basis, the proof will be complete if we are able to show that for ǫ small enough, under P n , trees t are typically close, in the Gromov-Hausdorff sense, to t[⌊ǫn⌋] in a uniform way as n → ∞. This is what the next lemma is taking care of.

Lemma 22

For every η > 0, one has

Proof. Conditionally on its skeleton Sk ⌊ǫn⌋ = (×, x, y), and conditionally on the set G ⌊ǫn⌋ of probability going to 1, the whole tree is reconstructed by grafting trees all with sizes ≤ ⌊ǫn⌋ to the tree Π -1 0 (×, x), as in Lemma 6. Let r 1 , r 2 , . . . be these sizes, in decreasing order, and t 1 , t 2 , . . . be the associated trees (with some arbitrary convention for ties). Of course, i r i = n. Conditionally on the sizes, the trees are random elements, respectively with distribution P r i , by Lemma 6.

At this point, we rely on the following particular case of [


Vincent Zoma
email: vincentzoma3@gmail.com

Wendpanga Jacques

Ismaël Tarama

Local community and use of Ceiba petandra in Banwa province in Burkina Faso 1

Keywords: Ceiba Pentandra, communities, Banwa, Burkina Faso

Regardless of the continent, man has a relationship with plants, but each community ritualizes its

relationship with its environment according to various factors. This relationship is often distinct from the practices of other neighboring communities and influenced by the language of communication. This research aims to analyze the relationship between sociolinguistic communities of the Banwa province in the western part of Burkina Faso, and the species Ceiba pentandra (L.) Gaertn. The fieldwork based on an ethnobotanical survey involved 267 people chosen at random and divided into three age groups comprising men and women, most of whom are illiterate. It essentially emerges from the study that the values of the indices of diversity and equitability of knowledge are low according to gender, and according to age. These values indicate an unequal distribution of knowledge of the species between respondents. In addition, the study revealed 6 forms of uses of the products of the species and among the different parts of the species, the leaves are more in demand than the bark and the fruit.

INTRODUCTION

Humans use elements of the environment such as plants for survival. Indeed, as noted by [START_REF] Bascot | 10 daily uses of plants you never knew existed[END_REF], the plant world that surrounds us and feeds us never ceases to amaze us because man has learned to use what nature puts in his hands, in particular, flowers, leaves, bark, sap, etc. for millennia, not only to eat, but also to heal, clothe and house.

Regardless of the continent, man has a relationship with plants ( [START_REF] Stéphanie | Anthropology of the relationship of Man to Nature: Nature experienced between destructive fear and intimate communion[END_REF], [START_REF] Danet | Interactions between plants in a community context: an experimental approach in Spain and Bolivia[END_REF]). As the [START_REF]the threatened flora of metropolitan France, online[END_REF], we live in plants.

Just look around us, or around cities, and the observation is there: the landscape is made of plants, they populate our plates, make and retain our soils, adorn our lives and our walks. Thus, each community ritualizes its relationship with its environment according to various factors. This relationship is often distinct from the practices of other neighboring communities and influenced by the language of communication [START_REF] Kelbessa | The rehabilitation of traditional environmental ethics in Africa Diogenes n° 207[END_REF].

It is in this context that the present study aims to analyze the relationship between sociolinguistic communities of the Banwa province in the western part of Burkina Faso, and a plant species (Ceiba pentandra (L.) Gaertn).

The species is known to be useful to humans. Indeed, Ceiba.pentandra (L.) Gaertn called kapok or cheesemaker in French, is a species well known to certain local communities for its various uses (in traditional medicine, in food, as a source of energy for cooking, in the field of crafts, etc.). The cheese maker has a sacred meaning for local populations in many parts of the world, including in tropical Africa where it often serves as a palaver tree ( [START_REF] Siepel | Ecological Profiles of large timber species[END_REF], [START_REF] Burkill | The Useful Plants of West Africa[END_REF], [START_REF] Irvine | Woody Plants of Ghana with Special Reference to their Uses[END_REF], [START_REF] Dick | Extreme long-distance dispersal of the lowland tropical rainforest tree Ceiba pentandra L. (Malvaceae) in Africa and the Neotropics[END_REF], [START_REF] Lobo | Effects of pollination by bats on the mating system of Ceiba pentandra (Bombacaceae) populations in two tropical life zones in Costa Rica[END_REF]). At the level of the natural range of Ceiba pentandra (L.) Gaertn, a large number of studies relate to its taxonomy ( [START_REF] Guinko | Illustrated flora of the Kou classified forest[END_REF], [START_REF] Eyog | The eatable Forest Fruit Trees of Cameroon[END_REF]). However, studies devoted to evaluating the ethnobotanical knowledge of the species and its woody potential are rare.

The present study therefore aims, as we have already mentioned, to analyze the relationship that exists between sociolinguistic communities of the Banwa province in the western part of Burkina Faso, and the species Ceiba pentandra (L.) Gaertn.

METHODOLOGICAL APPROACH

The study area of this investigation is the Banwa River. This province is largely a flat area. The relief is quite monotonous and altitudes rarely exceed 360m [START_REF] Kissou | Morpholopedological study of the Kossi and Banwa provinces, scale 1/100,000[END_REF].

For data collection in the field, a semi-structured survey questionnaire was developed. Before its use, a test made it possible to assess its applicability in the field. This contributed to better planning of the field data collection process. Then it was administered to the populations to collect information on the uses of the species. The main data collected relate to the identity of the respondent, the respondent's knowledge of the species and its uses.

Two categories of ethnobotanical studies were used.

These are the descriptive ethnobotanical study and the causal ethnobotanical study. These two approaches are complementary in the sense that they make it possible to answer different specific questions in a given area [START_REF] Albuquerque | Five problems in current ethnobotanical research and some suggestions for streng theming them[END_REF].

Regarding sampling, the random method was used.

It consisted in selecting within the surveyed population, a sample of size having knowledge on the use of the species so that each element of the sample has the same probability of being selected and that all elements of the population have the same chance of belonging to the sample ( [START_REF] Cochran William | Sampling techniques[END_REF], [START_REF] Scheaffer | Elementary Survey Sampling, 7thedn[END_REF]). The following mathematical formula established by [START_REF] Dagnelie | Theoretical and applied statistics[END_REF] was used to estimate the size of the sample to be considered:

𝑛 = 𝑈 1-𝛼 2 2 × P(1 -p) 𝑑 2
 n: the size of the sample to be considered.

 p: the proportion of informants who declared having knowledge about the use of the plant. This proportion p of informants was determined a posteriori.

  d: this is the margin of error of the estimate which has been set at a value according to the desired precision; its value can be between 5% and 10%.

In the context of this study, it was set at 5%. The method thus adopted has already been used in several ethnobotanical studies in West Africa ( [START_REF] Dadjo | Uses and management of black plum (Vitex doniana Sweet) in Southern Benin[END_REF], [START_REF] Koura | Ethnic differences in use values and use patterns of Parkia biglobosa in Northern Benin[END_REF], [START_REF] Assogbadjo | Biodiversity and socioeconomic factors supporting farmers'choice of wild edible trees in the agroforestry systems of Benin (West Africa)[END_REF]).

For the choice of people to be surveyed during data collection, the so-called "snowball effect" method was applied. It consisted of initially identifying a competent informant for the subject of study and then the latter after his passage of the survey, in turn indicates another competent informant from the same community. This process continued until the investigation of all the informants competent for the subject of the study in each locality concerned. At the end of the survey, the size of the sample of informants was 267 people. Given this number, the application of the previous formula made it possible to determine the proportion p=0.21.

Concerning the analysis, the survey data were summarized in tables using the EXCEL 2010 software. After the data entry operations, the same software was used to generate the graphs illustrating the results obtained. To assess the uses and importance of Ceiba pentandra (L.) Gaertn according to ethnic groups, three statistical parameters were calculated:

 The Diversity Index (DI) of respondents'

knowledge. [START_REF] Shannon | The mathematical theory of communication[END_REF] ecological diversity index was used to estimate the diversity of uses of Ceiba pentandra (L.) Gaertn. This index was calculated using the following formula:

𝐼𝐷 = -∑ ( 𝑁𝑖 𝑁 ) 𝑙𝑜𝑔2( 𝑁𝑖 𝑁 )
Ni= is the number of uses cited by the respondent and N is the total number of uses recorded by the survey.

-If all respondents have the same knowledge of the species, the value of the index is maximum and equal to DImax=Log2n

where log designates the natural logarithm and n is the total number of respondents.

Diversity is low when DI≤3; average if DI is between 3 and 4; then raised when DI≥4.

-The Fairness Index (EI) of knowledge of respondents. By equity index, we mean the ratio between real diversity and maximum theoretical diversity [START_REF] Dajoz | Summary of ecology[END_REF]. According to the same source, this index is calculated from the following formula:

𝐸𝐼 = 𝐷𝐼/𝐷𝐼𝑚𝑎𝑥

DImax=log2n EI= Index of equitability of knowledge of respondents DI=Diversity index of knowledge of respondents Max DI=Maximum Knowledge Diversity Index

The EI varies between [0; 1] and makes it possible to measure the degree of homogeneity of the knowledge of the respondents. If IE ˂0.5 the diversity of respondents' knowledge is not homogeneous but if IE≥0.5 this diversity is homogeneous; this means an equitable distribution of knowledge within the populations surveyed for the use of the species.

These indices were calculated according to the gender and age groups of the population surveyed.

The age classes proposed by [START_REF] Assogbadjo | Folk classification, perception, and preferences of baobab products in West Africa: Consequences for species conservation and improvement[END_REF] were used.

DImax=log2n

EI= Index of equitability of knowledge of respondents DI=Diversity index of knowledge of respondents Max DI=Maximum Knowledge Diversity Index

The EI varies between [0; 1] and makes it possible to measure the degree of homogeneity of the knowledge of the respondents. If EI ˂0.5 the diversity of respondents' knowledge is not homogeneous but if EI≥0.5 this diversity is homogeneous; this means an equitable distribution of knowledge within the populations surveyed for the use of the species.

These indices were calculated according to the gender and age groups of the population surveyed.

The age classes proposed by [START_REF] Assogbadjo | Folk classification, perception, and preferences of baobab products in West Africa: Consequences for species conservation and improvement[END_REF] were used 𝑅𝐶𝐹 = 𝑆/𝑁 × 100 -S: the number of people who provided an answer in relation to a given use of products from the species studied.

-N: total number of people interviewed.

--The Use Value of [START_REF] Phillips | Les plantes utiles de Tambopata, Pérou. II: Tests d'hypothèses supplémentaires en ethnobotanique quantitative[END_REF] modified by [START_REF] Rossato | Ethnobotany of caiçaras of the Atlantic Forest coast (Brazil)[END_REF].

This index makes it possible to evaluate the importance of a plant in a community of populations surveyed. The use value of each organ was calculated by the said formula which is:

𝑈𝑉 = ∑ 𝑈𝑖/𝑛 𝐼𝑛 𝑖=1
Ui = the number of uses that each respondent knows per organ of the species studied.

n= the total number of informants interviewed.

RESEARCH RESULTS

The results of the research presented in this study relate respectively to the age structure and distribution according to the ethnic groups of the people surveyed, the ethnobotanical knowledge of the populations on the species and the product uses of Ceiba pentandra (L.) Gaertn.

Age structure and distribution according to the ethnic groups of the people surveyed

The number of people surveyed is 267 people, 33%

of whom are women and 77% men. These respondents belong to the following ethnic groups:

Bobofing, Bobowulé or Bwaba, Bobodioula, Peulh, Mossi. They have an average age of 57 years old and 89.51% of them are illiterate. In addition to the representatives of local authorities and forest services, the people surveyed are divided into several socio-professional categories. These are traditional leaders, blacksmiths, breeders, traditional healers and religious leaders.

In addition to the results related to the age structure and the distribution according to the ethnic groups of the people surveyed, the study deals with the ethnobotanical knowledge of the populations on the species.

Ethnobotanical knowledge of the populations on the species

The results (Table 1) show that the value obtained (DI = 0.44) for men is higher than that for women (DI = 0.23). Knowledge about the use of the species is considerable for men (EI=0.18) compared to women (EI=0.1). The overall values of diversity of use of the species studied as well as those of the equitability of knowledge on the use of the species according to sex are respectively 0.67<0.5.

Table 2 allows us to know the local names of the space according to the ethnic groups in our research area. Apart from taking into account the ethnobotanical knowledge of the populations on the species, the study addresses the uses of the products of Ceiba pentandra (L.) Gaertn.

Uses of Ceiba pentandra (L.) Gaertn products

The uses made of the products of Ceiba pentandra (L.) Gaertn, by the local populations are diverse and varied (figure 1). The results of the study presented in this part are the subject of a discussion in relation to the literature review in the rest of the study.

DISCUSSION OF THE RESEARCH RESULTS

The discussion of the results of the study concerns on the one hand, the diversity of knowledge on the species, and on the other hand, the health and food orientation of the uses of the species studied.

Varieties of knowledge about the species

The values of the indices of diversity and equitability of community knowledge of the species are low.

This indicates an unequal distribution of knowledge among respondents. Indeed, a small group of respondents retain most of the knowledge about the species. The values of the diversity and equitability indices reveal that men have more knowledge about the species than women. Thus, knowledge of Ceiba pentandra (L.) Gaertn in the study area varies widely according to gender and age.

These results are comparable to those of [START_REF] Akouêhou | Sociocultural importance of Artocarpus altilis (Parkinson) Fosberg (Moraceae) in southern Benin[END_REF], by [START_REF] Wedjangnon | Ethnobotanical characteristics and sociocultural importance of Mansonia altissima A. Chev. in Benin[END_REF] and [START_REF] Lougbegnon | Quantitative ethnobotany of the use of Chrysophyllum albidum G. Don by local populations in Benin[END_REF] obtained with other species that have been the subject of similar studies. These are respectively Artocarpus altilis, Mansonia altissina and Chrysophyllum albidum. For example, [START_REF] Akouêhou | Sociocultural importance of Artocarpus altilis (Parkinson) Fosberg (Moraceae) in southern Benin[END_REF] As for the local names of Ceiba pentandra (L.)

Gaertn, each ethnic group gives at least one name to designate the species. These names symbolize many realities and assume the existence of a long history with the species in the study area. The literature reports that other localities take their name from plants. The case of Vudome or Hountigome, a locality in the Ouatchi environment whose name means "under Ceiba pentandra (L.) Gaertn." [START_REF] Akpavi | Socio-cultural value of food plants: a preservation factor[END_REF]. This is also the case of the village of Bananrodougou (village of cheesemakers) located in the municipality of Bama, about twenty kilometers from Bobo-Dioulasso in Burkina Faso.

Health and food orientation of uses

The uses made of Ceiba pentandra (L.) Gaertn by the populations in the province of Banwa are diverse.

All the interviewees report the food use of the species. This may be linked to the interest in this use.

On this point, [START_REF] Eyog | The eatable Forest Fruit Trees of Cameroon[END_REF] report similar findings from this food use. These authors report the use of the leaves of the species in cooking.

As for the types of diseases treated by the species, they are varied in the study area. This could be related to its spectrum of use and for the diversity of its parts taken for medicinal purposes. Moreover, several of these diseases (fever, mental illnesses, conjunctivitis, abscess, paronychia, cough, malaria, diarrhea, hemorrhoids, female sterility) are cured by the plant as mentioned by [START_REF] Michel | Trees, shrubs and lianas of the dry zones of West Africa[END_REF] and [START_REF] Okafor | Identification, use and conservation of plants in southeastern Nigeria[END_REF]. However, the literature indicates other diseases cured by parts of the species and which are not revealed by the present study. According to [START_REF] Zerbo | Attribution to the knowledge of medicinal plants used in infant care in San country, Burkina Faso[END_REF], the plant treats growth retardation in children. Also, studies have listed Ceiba pentandra (L.) Gaertn as a plant used by the population in Congo Brazzaville in the treatment of gastroduodenal ulcers [START_REF] Ibara | Ethnobotanic survey about congolese medecinal plants presumed anti-ulcer[END_REF].

Regarding ethnobotanical use values, leaves and bark are used more. This could be explained by the preference of the organs sought. This result is confirmed by [START_REF] Okafor | Identification, use and conservation of plants in southeastern Nigeria[END_REF] who reveal that the leaves of Ceiba pentandra (L) Gaertn are used as vegetables in cooking. The same source reports that the bark is galactagogue, febrifuge and aphrodisiac. It can treat female sterility, cough, malaria, diarrhea, gastralgia, dental caries, gingivitis, etc.

These results presented above show that Ceiba pentandra (L) Gaertn is a very useful plant with regard to its wide range of uses.

CONCLUSION

This study on the use of Ceiba pentandra (L.) Gaertn by the sociolinguistic communities of the province of Banwa, in Burkina Faso reveals the interest that these groups give to the species.

Knowledge of the species is linked to the profile of the person interviewed. Thus, people of advanced age, especially men, know the species. This shows the work that needs to be done so that young people also know about the species and especially its virtues. Indeed, the study also shows that Ceiba pentandra (L.) Gaertn is used by sociolinguistic communities for food, health care, commerce, crafts, energy needs and construction. These different solicitations concern organs of the species. These are the leaves, barks, fruits, seeds, wood, stems, thorns, roots and flowers. The use of certain organs such as flowers, fruits (pods) and seeds can be harmful to the survival of the species if their harvest is not accompanied by precaution. Consequently, the people met are in favor of the rehabilitation of Ceiba pentandra (L.) Gaertn in the Banwa province because of its importance on the socio-cultural, socio-economic and ecological levels.

  province located in the administrative region of Boucle du Mouhoun in Burkina Faso (Map 1). It is part of the country's cotton zone. Map 1: Geographic location of the study area Banwa Province is entirely located in the South Sudanese climatic sub-zone of Burkina Faso. There are 2 seasons there. The rainy season from May to October, with average monthly temperatures varying between 25° and 28°C and a dry season from November to April marked by the harmattan. It is characterized on the one hand by a dry and cold wind with low temperatures (around 25°) from November to the end of January and on the other hand, by a dry and hot wind with high average monthly temperatures often exceeding 40° C from March to April. Precipitation, which varies from 700 to 1000 mm per year, is irregular and poorly distributed in space. Hydrographically, the province of Banwa has a very dense hydrographic network, especially in its western half. The hydrographic network falls within the watersheds of the Mouhoun River and the Kossi

Figure 1 : 2

 12 Figure 1: Proportions of responses related to the different categories of use of Ceiba pentandra (L.) Gaertn products in Banwa province

Figure 2 :Photo 2 :

 22 Figure 2: Frequencies of citations by respondents of organs of Ceiba pentandra (L.) Gaertn

  revealed that informants know about Artocarpus altilis with an unequal distribution of knowledge from a gender perspective (DI and EI men=0.05 and 0.53 versus DI and EI women=0.04 and 0.49).Moreover, in the study area, the elderly (age >60 years) have more knowledge of the species, as evidenced by the values of the relatively high diversity and equitability indices. This knowledge could be explained by the long experience accumulated on the plant by the elderly. According to[START_REF] Bakiri | Ethnobotanical investigation of a medicinal plant Peganum harmalal in the region of M'sila (Algeria) / Revue Agriculture[END_REF], the experience accumulated with age is the main source of information at the local level about the use of plants.

Table 1 :

 1 Diversity Indices (DI) and Equitability Indices (EI) of knowledge on the use of the species by respondents

	Sex	DI	EI
	Men	0,44	0,18
	Women	0,23	0,1
	Global	0,67	0,28
	Age
	Youth (≤30years)	0,16	0,07
	Adult (31-60 years)	0,22	0,09
	Source : study data

Table 2 :

 2 Local names of Ceiba pentandra (L.) Gaertn according to ethnic groups in the study area

	Ethnic groups	Local names	Meaning
	Bobodioula	Prêe-Prôo	fast growing tree
	Bobofing	Prêe-Prôo	fast growing tree
	Bobowulé	Djâa	Big tree
	Bwaba	Tchâamou	Tree indicating a village by its
			height
	Bwaba	Tchônou	Big tree
	Dafing	Banan yiiri	Tree that cures many diseases
	Dafing	Bamba	Big tree
	Mossi	Gunga
	Peulh	Bâtinée