K Ram

Mohan Rao

Corinne Nouveau

K Trinadh

Mohan K Ram

Rao

ICAMEES2018 Low Temperature Plasma Nitriding of Low Alloy Steel for the Enhancement of Hardness

Keywords: Plasma nitriding, low alloy steel, hardness, X-ray diffraction, case depth

The present concerns surface modification of low alloy CrMoV steel by following the route of plasma nitriding for the improvement of hardness. Plasma nitriding was performed at a low temperature of 450 o C. The maximum hardness at this lower temperature was found to be ~1270 Hv after nitriding. At 550 o C though the hardness improvement near the surface region was similar but the core hardness was reduced. XRD studies revealed various nitrides of iron (γ′ (Fe 4 N) and ε (Fe 2-3 N)) with the ε nitride as the dominant phase. SEM analyses revealed the surface microstructure with almost no white layer. EDS point analyses shown the signature of N on the surface layer. It has been concluded in this study that the hardness of 90CrMoV8 steel can be improved significantly even at a low temperature of 450 o C without any of risk of the loss of core hardness.

Introduction

In recent past, the cutting tool industries were much concerned about the wear and corrosion resistance properties of the tool surface to prolong the service life of the cutting tools. Hard coatings improve the wear and corrosion resistance of cutting tools but the poor adhesion and delamination limit its application [1,2]. Instead of coating surface with the hard layer, if the surface is alloyed the risk of poor adhesion and delamination can be eliminated. Nitriding of the surface was found to be a successful process for the improvement of these properties. It has also been realized that nitriding followed by the coating improves the adhesion of the coated layer [3,4]. Conventional nitriding uses ammonia with no controllability of the process to achieve the desired properties or the modification of the microstructure. Plasma nitriding as one of the most industrially accepted and eco friendly plasma based processes which has successfully enhanced these properties of steels with a good controllability of the process parameters which enable the surface microstructure modified to desired properties [5][6][7][8][9][10].

Previously, plasma nitriding of various steels have been studied, however less is known about plasma nitriding of low alloy 90CrMoV8 steel [11][12]. In this study, attempts had been made to secure a high hardness at a low temperature of 450 o C and thus eliminating the higher temperature treatment to avoid the risk of distortion of surface or bulk properties. For the sake of comparison nitriding at higher temperature of 550 o C has also been attempted in this study.

X-ray diffraction, scanning electron microscopy (SEM) and energy dispersive spectroscopic analyses (EDS) have been made to understand the effects on surface microstructure of the post nitrided steel.

Following the structural characterization, hardness was measured by Vicker's micro hardness tester.

It has been concluded that the low temperature nitriding can also improve the hardness and case depth significantly.

Experimental

Samples with dimensions 10x10x3 mm 3 were cut from a sheet with the composition as given in following Table 1. The sample coupons were then subjected to metallographic polishing followed by ultrasonic cleaning in acetone. The cleaned sample coupons were then placed on the sample holder inside the nitriding reactor and evacuated the chamber. Initially, the chamber pressure was kept at 0.5 Pa and then increased to a fixed working pressure of 500 Pa. The process started with the Ar+ sputtering to clean the surface layer e.g. native oxide layer or other deposits until the desired temperature was attained.

Once the nitriding temperature 450 o C was achieved sputtering was stopped and nitrding cycle initiated. In the glow discharge plasma of N 2 and H 2 (at the ratio 80:20) nitriding was performed for 6h by varying the temperature. The nitriding cycle was closed after nitriding and the cooling cycle in the furnace environment began. All these process cycles were repeated again for nitriding at 550 o C (higher) temperature for 6h. The sample holder was biased negatively at 250V.

Nitrided sample coupons were exposed toscanning electron microscopic(SEM-modelSEM-Jeol JSM-5900/EDS) analyses. For the microstructural evaluation samples were exposed to X-ray diffraction(XRD -INEL CPS 120 diffractometer-θ/2θconfiguration)using Co-k α (0.17932 nm) radiation. Hardness measurements were performed by Vicker's micro hardness tester(LECO MHT-210 microhardness tester) at an applied load of 50 g. Each test was repeated ten times and the average of these hardness values was reported.

Results and analyses

Microhardnss measurements

Micro hardness measurements of the post nitrided samples were taken from the edge towards the bulk of the samples (Fig. 1a andb). Fig. 1 shows the hardness vs. depth profile of steels nitided at a lower temperature of 450 o C as well as at the higher temperature of 550 o C.The maximum hardness achieved was 1270 H v . This is a ~ 2 fold improvement in hardness which is a significantly higher than that shown by untreated steel .

The maximum hardness obtained after nitriding at the lower temperature 450 o C as well as at the higher temperature 550 o C was found to be ~1270 H v , however the case depth was much shallower at lower temperature nitriding i.e. ~50 µm. Case depth was considered to be an increase of 10 % over the core hardness. At higher temperature 550 o C the case depth was found to be ~ 120 µm. It should be noted here that this increase in case depth is much more in this case than that obtained at 450 o C whereas the maximum hardness on the surface remained almost unaltered. A decrease in case depth in the latter case may be attributed to the preferential diffusion of N than the formation of nitrides in the surface region. The increase in hardness and also the case depth at higher temperature is significant with a marginal loss of core hardness by only ~50 H v . Hence, nitriding at a higher temperature 550 o C for a shorter duration of 6h may be a good choice for achieving a wider hardened layer if the core loss of ~50 H v is acceptable. This will prolong the life of the tool in the service condition as the wear resistance for a wider case depth is expected to be more.

Fig From the above table, it is evident that the plasma nitriding at elevated temperature raises the hardness of CrMoV steel significantly to 1270 H v .

The studies so far were concentrated mainly on high temperature nitriding at 500 o C however, least is known about low temperature nitriding of this steel. In the present study,nitriding was performed at a lower temperature of 450 o C as well asat the elevated temperature of 550 o C. At this lower temperature also the hardness achieved was 1270 H v . This is to be noted here that Pokornýet al. [START_REF] Zdeněk | Tereza Mrárázková and Lacslo Fecso[END_REF] shown an increase in hardness to 1100 H v but at the loss of core hardness ~80 H v whereas in the present study the hardness achieved was more than this hardness without the loss of core hardness.

This increase in hardness in the present study could be attributed to the higher concentration of Cr which is 8% whereas in the other cases it is around 3% only. In any case, the present study shows an achievement in hardness much higher than obtained by other workers so far.

Fig. 1 shows that nitriding at lower temperature of 450 o C increases the maximum hardness 1270 H v which is the same as obtained after nitriding at elevated temperatures as shown by others workers [16,17] and also as shown in in the present study at 550 o C.

X-ray diffraction and SEM analyses

For the phase evolution nitrided steels treated at various temperatures were subjected to detailed X-ray diffraction (XRD) analyses. increased. These phases were found to be stable even at higher temperature treatment. The total peak areas of these ε phases seem to be greater than that of γ' phase.

Cross sections of the post nitrided samples were polished and etched with the Villela's reagent. Fig. 3 represents SEM and EDS analyses of one of the selected sample treated at 450 o C for 6h revealing the absence of white layer on the surface which is advantageous for the integrity of mechanical properties. EDS analyses across the cross section have been done at different points from surface to bulk in all the nitrided steel samples and a representative micrograph and EDS analyses are shown in the above Fig. 3. EDS reveals the signature of nitrogen in the nitrided steel along with other elements. It was also found that the concentration of nitrogen decreases from surface to bulk. This is in accordance with the decreasing hardness from the surface to bulk.

Conclusions

Plasma nitriding of low alloy CrMoV steel has shown the improvement of hardness significantly when treated for a shorter duration of 6h and at also at a lower temperature of 450 o C. The hardness achieved was ~1270 H v . Nitriding at 550 o C for 6 h has also increased the hardness to ~1270 H v without any significant loss of core hardness. At 550 o C nitriding, the case depth increased was to ~ 120 µm which is significantly greater than achieved at 450 o C. Thus, nitriding for 6 h at this temperature is safe for achieving a higher hardness with a wider case depth without affecting much to the loss of core hardness.

Fig. 2

 2 reveals the XRD profiles of samples nitrided at 450 o C and 550 o C for 6h along with the as-received sample.

Fig. 2

 2 Fig. 2 Bragg-Brentano mode of X-ray diffraction profiles of as-received and nitrided CrMoV steels at variable temperatures (450 and 550 o C) for a fixed duration of 6h.

  23 o a prominent peak of γ' (111) appears and another peak γ' (200) with less intensity appears at 54.9 o . ε (100), ε (110) and ε (103) peaks at 44.6 o , 82.5 o and 91.4 o respectively were grown at lower temperature 450 o C. At this temperature Fe (110) peak converted to ε (111) peak as the nitriding time

Fig. 3

 3 Fig.3 SEM and EDS (a-a1) of CrMoV steel after plasma nitriding at 450 o C for 6h.

Table 1 :

 1 Chemical composition of CrMoV steel

	Element	C	Si	Mn	Cr	Mo	V	Fe
	Wt. (%)	0.5	1.0	0.5	8.0	1.5	0.5	balance

Steel type (martensitic) Plasma nitriding at temperature/ time (h) Maximum hardness (H v )

				Year	References
	3% Cr-Mo-V	538 o C/ longer	~770	2000	Rolinski et al.
		time			[13]
	3.25Cr-Mo-V	510 o C /6h	900	2004	Chala et al. [14]
	2.95% Cr-Mo-V	500 o C/6h	1100	2011	Pokorný et al.

.1 Microhardness vs. depth profiles of samples treated at 450 o C and 550 o C for 6 h.

Plasma nitriding at lower as well as elevated temperatures raises the hardness of CrMoV steel significantly. Hardness achieved in the present study is much higher than the previously reported hardness values

[13][14][START_REF] Zdeněk | Tereza Mrárázková and Lacslo Fecso[END_REF]

. Hardness values achieved by elevated temperature plasma nitriding of CrMoV steel by other workers are shown in the following Table

1

.

Table2. Maximum hardness achieved after plasma nitriding of CrMoV steels

Acknowledgements

The opportunity given to carry out this research and granting the permission to submit the work for the publication by GITAM (Deemed to be University), Visakhapatnam,Andhra Pradesh, India, is gratefully acknowledged. The authors also gratefully acknowledge the group members of ENSAM, Paris Tech., France for their valuable discussion and support to carry out this work.


Pablo Domínguez

Pastoral

Communities Ecosystem

populations through community governance system implying that access to and use of these ecosystems have to follow a series of rules set up by the local communities themselves (e.g. prohibition to graze during the most sensitive moments of the year like in spring), in order to:

-Give a rest to the plants in their key periods of growth and/or reproduction.

-Maximize annual production.

-Ensure the continuity of the ecosystems.

-Make human activity sustainable while benefiting communities and biodiversity.

Titre partie

Pastoral Commons: a tale of beauty and destruction

HIGHLANDERS

Traditional mountain communities are the first to be interested in conserving the ecosystems and natural resources on which their survival depends and to which they are deeply culturally connected. This mutual interest they share, guarantees greater conservation outputs and diminishes agro-territorial conflicts. Pastoral Commons: a tale of beauty and destruction

Communities

ECOSYSTEMIC ROLE

From a minimum (left) to a maximum (right) of the communal management system, we can observe the increase in vegetation cover (upper graph), in species and in families (lower graph). At the same time, among others, pastoral commons favor:

-Pools for seed conservation and diffusion.

-Soil retention against erosion.

-Availability of water.

-Carbon sinks against climate change.

-Diversity of landscapes.

Communities Ecosystem

Needs Sinjajevina

URGENT NEEDS

It can be assumed that there are hundreds of thousands of commons in the Mediterranean region and that they could cover more than half a million km2.

But it's impossible to know their exact range, as well as their precise global environmental, socio-cultural and economic role, unless a major comparative research on these is launched.

This would also be the first step towards elaborating an effective protection and promotion plan that can no longer be postponed, since most of these systems are threatened or in degradation phases. In fact, all over the Mediterranean and for many centuries up to our own days, abusing them is a generalized pattern that must be reverted.

THE EXAMPLE OF SINJAJEVINA

Sinjajevina is a cluster of pastoral commons in Montenegro, the biggest mountain pasture of the Balkans, the second in Europe and a vital ecosystem against climate change and in favor of biodiversity conservation. People here have not forgotten they are one with the land they live in. They synchronize with the rhythm of nature and nature with them. Around them, undegradated slopes reach as far as the eyes can see, unbearably green in summer, unbearably white in winter. In many senses a heavenly place. But as it always happens, some see heaven and feel like eating the apple.

Pastoral Commons: a tale of beauty and destruction

Communities Ecosystem Needs Sinjajevina

In 2019 the government of Montenegro supported by NATO, decided to inaugurate a military training ground in Sinjajevina without measures against the damage this would represent to the land, to its people and to these socio-ecologically invaluable commons.

Once we destroy these systems, it is very hard (to not say impossible), to reconstruct their thousands of years' living heritage, so we should think on protecting them rather than destroying them.

This military training ground stands as an example of the generalized state of constant threat on most commons in the Mediterranean basin, as they are rarely legally recognized, and of how urgent it is to address them and to work to protect them at the Mediterranean scale.

Saving Sinjajevina is a first step towards preserving all Mediterranean pastoral communal ecosystems. And it's one worth taking.

Pastoral Commons: a tale of beauty and destruction

Communities Ecosystem Needs Sinjajevina

  a tale of beauty and destruction Titre partie MEDITERRANEAN MOUNTAIN PASTORAL COMMONS Pastoral commons are at the same time natural and cultural. They are pasturelands conserved by local

  pastoral commons in the Mediterranean within the current research gap.


Pascal Ferraro
email: pascal.ferraro@labri.fr

Christophe Godin
email: christophe.godin@cirad.fr

Przemyslaw Prusinkiewicz

Toward a quantification of self-similarity in plants

Keywords:

Self-similarity of plants has attracted the attention of biologists for at least 50 years, yet its formal treatment is rare, and no measure for quantifying the degree of self-similarity currently exists. We propose a formal definition and measures of self-similarity, tailored to branching plant structures. To evaluate self-similarity, we make use of an algorithm for computing topological distances between branching systems, developed in computer science. The formalism is illustrated using theoretical branching systems, and applied to analyze self-similarity in two sample plant structures: inflorescences of Syringa vulgaris (lilac) and shoots of Oryza sativa (rice).

INTRODUCTION

Repetitive patterns are readily noticeable in the growth and structure of many living organisms. In particular, modular organization is an essential element of the development and structure of plants. [START_REF] Hallé | Tropical trees and forests. An architectural analysis[END_REF][START_REF] White | [END_REF][START_REF] Harper | Population biology of plants[END_REF][START_REF] Barthélémy | [END_REF]5 This is essential to the understanding of plant biology, and plays an important role in the formal analysis 6 and simulation 7,[START_REF] Prusinkiewicz | The algorithmic beauty of plants[END_REF] of plants.

In some cases, the modules are arranged into compound, recursively nested, fractallike structures, with similar patterns appearing at different scales [START_REF] Mandelbrot | The fractal geometry of nature[END_REF] (Fig. 1). In botany, an early study of such structures is due to Arber. [START_REF] Arber | Natural philosophy of plant form[END_REF] Troll 11 defined a compound inflorescence as a system consisting of the main florescence and paracladia (term introduced originally by Schultz [START_REF] Schultz | Neues System der Morphologie der Pflanzen[END_REF] ) that repeat the structure of the main florescence. This concept was later formalized by Fritjers and Lindemayer, [START_REF] Frijters | Developmental descriptions of branching patterns with paracladial relationships[END_REF] who defined paracladia as "branches which repeat the florescence of the main axis and which on their turn can give rise to paracladia of their own." Prusinkiewicz et al. [START_REF] Prusinkiewicz | The use of positional information in the modeling of plants[END_REF] introduced a related concept of branch mapping, according to which, "given two branches of the same order, the shorter branch is identical [...] to the top portion of the longer branch." Mündermann [START_REF] Mündermann | Inverse modeling of plants[END_REF] further studied these concepts as a basis for constructing three-dimensional plant models. More recently, Prusinkiewicz [START_REF] Prusinkiewicz | Self-similarity in plants: Integrating mathematical and biological perspectives[END_REF] introduced the notion of topological self-similarity, which relates plant self-similarity to the theory of L-systems. [START_REF] Lindenmayer | [END_REF]18,[START_REF] Prusinkiewicz | The algorithmic beauty of plants[END_REF] In comparison to inflorescences, self-similar organization is less obvious in trees, which develop over a longer period of time, and therefore are more prone to the influences of the environment. Nevertheless, the branching systems of trees also result from repetitive processes, in which various meristems follow similar sequences of states and produce similar structures as a result. These sequences have been characterized by biologists in such terms as age state, 19 morphogenetic program, [START_REF] Nozeran | Integration of organismal development[END_REF] and physiological age. [START_REF] Barthélémy | Architecture, gradients morphogénétiques et âge physiologique chez les végétaux[END_REF] This last notion made it possible, for example, to exploit similarities in the development and structure of Zelkova serrata (Japanese elm) in the construction of a simulation model, [START_REF] De Reffye | Architecture et modélisation de l'orme du Japon Zelkova serrata (Thunb.) Makino (Ulmaceae): la notion d'axe de référence[END_REF] where the sequence of physiological ages of a typical meristem in the plant served as a template (called the reference axis) for all plant meristems. Self-similarity, like symmetry, is not easily quantifiable: it is usually considered to be either absent or present. Real plants, however, may be self-similar only to some degree. In this paper, we propose a definition of self-similarity in branching structures, and describe a procedure for quantifying it in measured or simulated plants. This procedure is based on a method for comparing tree-like structures introduced by Zhang [START_REF] Zhang | [END_REF] and extended to the problem of comparing plants. 24,25 We first analyze the self-similarity measures using branching structures with controlled, algorithmically generated topology. We then apply these measures to analyze self-similarity in two real plant structures obtained from field measurements.

THEORETICAL CONCEPTS 2.1 Self-Similarity of Axial Trees

A plant is viewed as an assembly of adjacent botanical components such as internodes or annual growth increments. [START_REF] Bell | Plant form, an illustrated guide to flowering morphology[END_REF] Such a structure can be formally described by defining a set of vertices V that represent the plant components (each vertex corresponding to a plant part: internode, growth unit, etc.), and a list E of pairs of vertices that describes the adjacency of these components. 6 We assume that each component v is physically attached to the plant body through at most one parent component, denoted parent(v).

The resulting topological structure is called a rooted tree graph T = (V, E). In such a graph, every vertex except one (the root r) has exactly one parent vertex: the root has no parent. In the following, a tree graph rooted in r will be denoted by T [r], and the empty tree graph will be denoted by θ = (∅, ∅). In order to identify the different axes on a given plant, two types of relations between entities are distinguished: an entity can either precede (symbol <) or bear (symbol +) another entity. An axial tree 8 is then defined as a rooted tree in which an entity can be attached to at most one other entity by a < connection. Formally, we have: Definition 1 (Axial tree) An axial tree is a graph T = (V, E, α), where V is a finite vertex set, E is a finite set of ordered vertex pairs, and α is a mapping of E into {<,+}, provided that:

• the edge type function α satisfies the condition:

if α(x, y) = α(x, z) = '<' then y = z for all (x, y) ∈ E and (x, z) ∈ E.
We define the axis of v as the set of vertices that are connected to the given vertex v through paths consisting only of '<'-edges. The second property of Definition 1 implies that this axis is always a linear sequence of vertices. We denote this sequence by A(v):

A(v) = {v 1 , v 2 , ..., v n } if v 1 = v and α(v i , v i+1 ) = '<' for all i = 1, 2, • • • , n -1.
(

) 1
We call the axis A(r) of the root r of T the trunk of T . For any vertex v, rank(v) is the number of vertices in A(v). These definitions are illustrated in Figs. 2.a,b.

Definition 2 (Axial tree isomorphism) Let us consider two axial trees,

T 1 = (V 1 , E 1 , α 1 )
and

T 2 = (V 2 , E 2 , α 2 ). A bijection φ from V 1 to V 2 is an axial tree isomorphism if:
• for each (x, y) ∈ E 1 , (φ(x), φ(y)) ∈ E 2 , and

• α 2 (φ(x), φ(y)) = α 1 (x, y).
Two structures T 1 and T 2 are thus isomorphic (denoted T 1 ≡ T 2 ) if they are identical except for the labels of their components. We can now formalize the notion of paracladia:

Definition 3 (Paracladium) Let v be a vertex of T . T [v] is a paracladium of T if there exists a vertex w in the trunk of T such that T [v] ≡ T [w].
The notion of paracladium enables us to define a notion of self-similarity which captures the main features of nested structures discussed in the introduction.

Definition 4 (Self-similarity) An axial tree T is self-similar if all of its sub-branching systems are paracladia, i.e. ∀v ∈ V , T [v] is a paracladium of T .

In a self-similar axial tree, the trunk provides a template for the branching pattern of the entire structure. A small branching system is isomorphic to a distal part of the trunk branching system, while a large branching structure is isomorphic to a larger distal part of the trunk branching system. Figure 3 illustrates this definition using different self-similar branching structures as examples. We note that self-similar structures may have different degrees of apparent structural complexity, related to their maximum branching order. Now, let us consider the question of deciding whether a given axial tree structure is self-similar. A naive approach based on Definition 4 would consist of checking, for each vertex v of the tree, whether there is a corresponding vertex w of the trunk such that T [v] ≡ T [w]. However, the following proposition enables us to greatly simplify this approach by making use of the recursive character of self-similar branching structures.

For any vertex v ∈ V, let us call B(v) the set of vertices x ∈ V , such that x ∈ A(v) and parent(x) ∈ A(v). B(r), for example, is the set of root vertices of all first-order branches of the tree T [r] (Fig. 2.c).

Proposition 5 An axial tree T [r] is self-similar if and only if for every vertex v ∈ B(r),

the branch T [v] is a paracladium of T [r].
The proof is given in the appendix. Its intuition is as follows. Suppose that all of the first-order branches have been proved to be paracladial. This means that any such branch B 1 is isomorphic to some distal part T 1 of the trunk (Figure 4.a and b). Any second-order branch B 2 of B 1 is then isomorphic to a branch T 2 of T 1 . However, a branch of T 1 is a first-order branch; hence, by hypothesis, T 2 is a paracladium, and so must be B 2 . By applying this argument recursively we show that any branch B n is a paracladium, because branch B n of order n is isomorphic to some branch B n-1 of order n -1, which in turn is isomorphic to some branch B n-2 of order n -2, and this chain of equalities will eventually stop at some distal portion of order 0 of the trunk branching system. Proposition 5 has two major implications. First, it shows that the isomorphism between first-order branches of a tree T and distal portions of the trunk of T is a necessary and sufficient condition for the self-similarity of T . Second, is reveals that nesting of paracladial structures is an inherent feature of self-similarity as specified by Definition 4:

the sub-branches of larger paracladial branches must themselves be paracladial branches.

We will make use of these properties in the following sections.

Comparing Branching Systems to Assess Self-Similarity

Let us now consider the computational aspect of verifying whether two axial trees are isomorphic. For simple trees, this problem can be solved by recursively comparing the branching systems borne by the trunk, starting at the root. In general, however, each node may bear several branches, which makes deciding whether two structures are isomorphic a difficult combinatorial problem.

Fortunately, this problem turns out to have an efficient algorithmic solution. In this approach, verifying whether two branching structures are identical comes down to counting the minimum number of atomic operations required to transform one structure into the other. If this number turns out to be 0, then the two structures are isomorphic. The method thus relies on the computation of edit distances between tree structures. [START_REF] Zhang | [END_REF]24 In this section, we sketch the basic principle underlying the definition of this distance, which will subsequently be used as the main mathematical tool for the definition of a measure of self-similarity.

The evaluation of similarity between branching structures has been studied in computer science and is known as the tree-to-tree comparison problem. [START_REF] Ohmori | [END_REF] The distance between two tree graphs is defined as the minimum cost of a sequence of edit operations which transforms one tree graph into the other. We consider three kinds of atomic edit operations on a tree graph T : 23 substituting one vertex for another (note that this changes their labels), deleting a vertex, and inserting a vertex. A constraint is added to the de-finition of insertions and deletions 1 : if a node is inserted between a parent node and its children, the new node must become the parent vertex of all the children (and not only of a subset of them); deletions are similarly constrained.

A cost function is defined for each edit operation s which assigns a non-negative real number c(s) to s as follows: c(s) = d sub (v, w) if s is a substitution of v by w; and c(s) = d indel (v) if s is an insertion or a deletion of vertex v. We assume, for any pair of vertices v and w, that

d sub (v, w) ≤ d indel (v) + d indel (w).
In this work, we use a purely topological elementary distance, 24 which captures differences in the arrangement of plant components without taking their properties (such as type or geometry) into account. Formally, d sub (v, w) = 0 and d indel (v) = 1 for any pair (v, w) of vertices of the tree graph. Using this elementary distance restricts the evaluation of self-similarity to the topology of plant architecture.

Let S = (s 1 , s 2 , ..., s n ) be a sequence of n edit operations that transforms a tree graph T 1 into another tree graph T 2 . The cost C(S) of S is defined by summing up the cost of the edit operations that compose S: C(S) = s∈S c(s i ). The dissimilarity measure D(T 1 , T 2 ) between a tree graph T 1 and a tree graph T 2 is then defined as the minimum cost of any sequence that transforms T 1 into T 2 . If this distance is 0, then no operations are required to transform T 1 into T 2 ; this only happens if T 1 is isomorphic to T 2 . It can be shown that this dissimilarity measure is actually a distance 232 . Due to the definition of this distance, the larger D(T 1 , T 2 ), the more different the structures T 1 and T 2 .

The computed distance strongly depends on the size of the compared tree graphs. In order to make the comparison results size-independent, we create the normalized dissimilarity measure D by dividing the distance by the total number of vertices in compared [START_REF] Hallé | Tropical trees and forests. An architectural analysis[END_REF] Zhang and Jiang 28 have shown that the computation of distance between tree graphs using unconstrained edit operations is a MAX SNP-hard problem, i.e., there is no polynomial-time solution or approximation scheme for this problem. The introduction of constraints makes it possible to find a solution in polynomial time.

2 That is, that

D(T, T ) = 0; D(T, U ) > 0 if T ≡ U ; D(T, U ) = D(U, T ); and D(T, U ) ≤ D(T, X) + D(X, U ).
tree graphs:

D(T 1 , T 2 ) = D(T 1 , T 2 ) |T 1 | + |T 2 | . Since D(T 1 , T 2 ) is a distance, D(T 1 , T 2 ) ≤ D(T 1 , θ) + D(θ, T 2 ) = |T 1 | + |T 2 |, which implies that D(T 1 , T 2 )
D(T 1 , T 2 ) = 0, i.e. T 1 ≡ T 2 .
Unlike D(T 1 , T 2 ), however, the normalized dissimilarity measure does not always satisfy the inequality D(T, U) ≤ D(T, X) + D(X, U) required for it to be a distance.

Distance Between Axial Trees

We adapt the notion of dissimilarity measure to quantify self-similarity between axial trees by constraining edit operations so that they maintain the integrity of axes. The need for this constraint is illustrated in Fig. 5.b, which shows two structures would be isomorphic if the axial information is not taken into account, but are not isomorphic if the mapping between nodes respects the macroscopic axis structure. Formally, if v 1 and w 1 are two vertices of an axial tree T 1 that are transformed respectively into vertices v 2 and w 2 in the axial tree T 2 , then the transformation should be such that:

v 1 ∈ A(w 1 ) ⇔ v 2 ∈ A(w 2 ). (2)
In other words, if v 1 and w 1 belong to the same axis in T 1 , then their images v 2 and w 2 should also belong to the same axis in T 2 . An algorithm for computing distances associated with such constrained transformations between quotiented trees3 was described by Ferraro and Godin. 25 Since axial trees are special cases of quotiented trees, this approach enables us to define a distance D A (T 1 , T 2 ) between axial trees. As for the distance between simple trees, a corresponding normalized dissimilarity measure D A (T 1 , T 2 ) between axial trees T 1 and T 2 can be defined such that:

0 ≤ D A (T 1 , T 2 ) ≤ 1, (3)
D A (T 1 , T 2 ) = 0 ⇔ T 1 ≡ T 2 . (4)
Here T 1 ≡ T 2 denotes an axial tree isomorphism, i.e. an isomorphism respecting the axis organization in both trees. This latter property and the existence of an algorithm for computing the measure D A (T 1 , T 2 ) provides us with an algorithmic tool for addressing the original question of verifying whether two axial trees are isomorphic.

Paracladial Coefficient of v

Let T be an axial tree. For any v in T , let us define ι(v), the paracladial image of v, as the vertex on the trunk such that rank(ι(v)) = rank(v).

Now, for a given vertex v of T , let us define the paracladial coefficient γ(v) as:

γ(v) = 1 -D A (T [v], T [ι(v)]) (5)
Note that the value of γ(v) is in the interval [0, 1]: it is close to 0 if the branch structure is very different from the trunk structure, and close to 1 when the branch T [v] is similar to the trunk. In particular, γ(v) is equal to 1 when the branch T [v] is isomorphic to the distal part of the trunk branching structure; that is, when it is a paracladium. We can thus restate Proposition 5 using the notion of the paracladial coefficient:

Corollary 6 The axial tree T is self-similar if and only if γ(v) = 1 for all v ∈ B(r).
This corollary states that self-similarity can be detected in an axial tree by computing

Approximate Self-Similarity

As mentioned in the introduction, real plants are usually self-similar only to some degree, if at all. It is thus necessary to study how the definition of pure self-similarity introduced above can be extended to account for approximate self-similarity.

Since the measure D A (T 1 , T 2 ) reflects a structural dissimilarity between the two axial trees T 1 and T 2 , it also reflects how far the two structures are from being isomorphic.

More precisely, this measure defines the percentage of changes (with respect to the size of the compared tree structures) that must be made to transform one structure into the other. This property of D A can be used to introduce a coefficient reflecting the average self-similar quality of a tree structure.

Definition 7 (Mean self-similarity coefficient, MSC) Let T be an axial tree rooted in r. The mean self-similarity coefficient γ(r) is the mean value of the paracladial coefficients γ(v), where v ranges over all the first-order branches of T [r]:

γ(r) = 1 |B(r)| v∈B(r) γ(v). (6)
For simplicity, we write γ(T ), or simply γ, when the argument T [r] is clear from the context. Note that the MSC γ(r) is equal to 1 if the tree T [r] is perfectly self-similar, and it is close to 0 if the tree is weakly self-similar (Fig. 6.c). Thus, the MSC can be used as a measure of self-similarity of plants.

Plants may have branches that vary greatly in size. For example, monopodial plants frequently have short axes at the top of the trunk and longer axes at the bottom. The probability that a branch resembles a distal part of the whole tree is higher for a short branch than for a long branch. Consequently, short branches tend to have high paracladial coefficients, which introduces a bias in the estimation of the degree of self-similarity (MSC)

of the whole tree. To compensate for this bias, we define weighted mean self-similarity, which assigns a higher weight to the longer branches.

Let L(r) be the length of all the axes borne by the trunk of T [r]:

L(r) = v∈B(r) rank(v). (7)
For each branch borne by the trunk of T [r], rooted in v ∈ B(r), we define weight α(v) as

α(v) = rank(v) L(r) . (8) Obviously, v∈B(r)
α(v) = 1. (9)
We then define the weighted paracladial coefficient at node v as

γ ′ (v) = α(v)γ(v). ( 10
)
Definition 8 (Weighted mean self-similarity coefficient, WMSC) Let T be an axial tree rooted in r. The weighted mean self-similarity coefficient γ(r) is the mean value of the weighted paracladial coefficients γ ′ (v), where v ranges over all the first-order branches of T [r]:

γ(r) = v∈B(r) γ ′ (v) = v∈B(r) α(v)γ(v). (11)
The notion of weighted mean self-similarity is consistent with Proposition 5, according to which a large, compound branch of a self-similar structure includes smaller paracladia as its parts. When evaluating the degree of self-similarity of a whole plant, the paracladial coefficients of the larger branches are thus more representative than the coefficients of the smaller branches, and therefore should carry more weight.

The mean self-similarity coefficient MSC and the weighted mean self-similarity coefficient WMSC characterize the average self-similarity of an entire plant. These characteristics can be complemented with the values of variance of the paracladial coefficients γ(v) and their weighted counterparts γ ′ (v), where v spans all the first-order branches of the plant. The resulting parameters, var v∈B(r) γ(v) and var v∈B(r) γ ′ (v), quantify how homogeneous a plant's structure is from the viewpoint of its self-similarity.

An even more detailed characterization can be obtained by listing paracladial coefficients of individual branches. The need for such a characterization is illustrated in Figs.

6.a and b, which contrasts two very different structures with a similar overall values of paracladial coefficients and their variances. In the structure of Fig. 6.a, the paracladial coefficients of the distal branches are equal to 1 (the branches are parcladial), whereas the paracladial coefficients of the basal branches are close to 0. The structure of Fig. 6.b has an opposite organization: its apical branches are not paracladial while its basal branches are. To discriminate between these cases, we need to analyze not only the average values of coefficients, but also their distribution along the plant axis. In general, if a quantity q(v) is defined for all positions v along the trunk, we call the data set (rank(v), q(v)) the profile of q(v) along the trunk. For example, a profile of γ(v) may show high values of γ near the apex and low values near the bottom of the trunk in a non-homogeneous case.

Applications of the numerical parameters and profiles to the analysis of plant structures are described the next section.

3 APPLICATION TO THEORETICAL SELF-SIMILAR PLANTS

Theoretical Plants

To evaluate the usefulness of the self-similarity parameters, we computed them for three families of algorithmically generated plant-like branching structures. The use of synthetic plants allowed us to precisely control the character of each structure and verify whether its self-similarity aspects are properly discriminated by our parameters.

The plants were generated using the L-system-based modeling program cpfg, [START_REF] Mech | Cpfg version 4.0 user's manual[END_REF] incorporated into the plant modeling software L-studio and vlab. [START_REF] Prusinkiewicz | [END_REF] Structure generation begins with a single shoot apical meristem and proceeds in a sequence of simulated developmental steps. In every step, the meristem adds a growth unit to the plant axis and recreates itself. Each growth unit supports a lateral apex, which may give rise to a next-order lateral axis. This process repeats for higher-order axes, resulting in the formation of a branching structure.

Following the notion of physiological age reviewed in the Introduction, we assumed that the apical meristem of the main axis progresses through a sequence of morphological differentiation states, from germination to the flowering state. The set of states is ordered, with the next state of an apical meristem in state s greater than or equal to s. The state of the apical meristem thus gradually increases until the apex reaches the final state and becomes a terminal organ (a flower). The lateral meristems follow a similar progression of states, with the initial state of each lateral meristem equal to or greater than the state of the apical meristem that has created it. See (Godin et al. 2005) for complementary details.

We visualize the above process using differentiation graphs that show the set of states and two types of possible transitions between them (Fig. 7). Colored circles represent differentiation states. Solid arrows represent possible state changes of the apical meristem during the apical growth of an axis. The meristem stays in the same state for the number of steps indicated by the label associated with a loop, then progresses to the next state.

For example, the differentiation graph of Fig. 7.a corresponds to the axis shown in Fig. for the final state). Model M2 differs from M1 in that some lateral meristems produced by the apical meristem in state s may assume state s ′ greater than s + 1. For example, the apical meristem in state 1 produces a lateral meristems in state 3. In model M3, a meristem in state s produces lateral meristems in state s ′ = s + 1, but there is no gradual progression of states along either the main or the lateral axes. Instead, after remaining in the same state for a number of steps, an apical meristem differentiates directly into a flower.

For each deterministic plant Mi, we generated a set of five derived plants, labeled Mi -0.8, Mi -0.6, . . . , Mi -0.0, by randomizing the functioning of the lateral meristems.

With probability p, a lateral apex of order greater than 2 gave rise to a branch; otherwise, the branch was aborted. This probability p is indicated in the plant name: p = 0.8 in Mi -0.8, p = 0.4 in Mi -0.4, and so on. First-order branching was not affected by this probability.

According to this design, the lateral branches of models M1 and M2 repeat parts of the main shoot structure; thus, structures M1 and M2 are self-similar in the sense of Definition 4 (Fig. 8). The random removal of branches in these structures introduces variation that is expected to reduce their degree of self-similarity. In contrast, the lateral branches of M3 are not copies of the main structure; thus, M3 is not self-similar, and its random variations are also not expected to be self-similar. Below we show how these qualitative characterizations are captured and quantified by the self-similarity parameters.

Analyzing and Comparing Self-Similarity of Theoretical Plants

The data were analyzed using the AMAPmod module for plant architecture analysis, 31 which incorporates algorithms for comparing tree graphs. 24 AMAPmod is a part of the ALEA modeling platform. [START_REF]ALEA: A software for integrating analysis and simulation tools for 3D architecture and ecophysiology[END_REF] The unweighted and weighted mean self-similarity coefficients, and the variance of unweighted and weighted paracladial coefficients, were computed for each plant of the three families M1, M2 and M3. The results are given in Table 1.

We observe that the MSC (mean self-similarity coefficient) of plants M1 and M2 is equal to 1. This indicates perfect self-similarity, which is consistent with the manner in which these plants were generated. In contrast, the MSC of plant M3 is equal to 0.4, which means that for each branch borne by the trunk, an average of 60 percent of its components should be added, removed, or rearranged to achieve the self-similarity condition of Proposition 5. This low value of the MSC is again consistent with the manner in which plant M3 was generated, since its simulation algorithm explicitly prevented the lateral branches from following the structure of their parent.

Within the randomized plant families M1 and M2, the reduction of the branching probability p is followed by a reduction in the value of MSC. This is consistent with our expectation that removing branches at random from an initially self-similar structure will decrease its self-similarity. In the family M3, in contrast, the overall decrease of the MSC from 0.4 for plant M3 to 0.3 for plant M3 -0.0 is not monotonic. This suggests that randomly removing branches in a non-self-similar plant may either increase or decrease its self-similarity.

Profile curves provide an additional characterization of plant architecture. The profile of the paracladial coefficients γ(v) for the reference plant M1 (top curve in Fig. 9.a)

shows that its branches are strictly paracladial: γ(v) = 1 for all positions v. This is consistent with the definition of M1 as a perfectly self-similar plant. In the randomized plants M1 -0.8 . . . , M1 -0.0, the edit distance between branches and the trunk tends to increase for branches positioned lower on the trunk. Although in Fig. 9.a this trend is obscured by the random variation of the paracladial coefficients γ(v), it is clearly visible in Fig. 9.b, which shows the self-similarity coefficients γ(r). The gradual decrease in the values of the paracladial coefficient and the self-similarity coefficient as rank(v) increases reflects the dependence of both coefficients on the size of branches: the lower branches in the plant family M1 are larger than the upper branches (Fig. 8, first row), and therefore more likely to depart from the trunk structure.

The weighted paracladial coefficients γ ′ (v) were introduced to compensate for this apparent overemphasis of the self-similarity of small branches. The values of γ ′ (v) for the plant family M1 are shown in Fig. 9.c; the lengths of branches that serve as weights are plotted in Fig. 9.d. As expected, we observe an increase in the value of γ ′ (v) proportional to the size of branches in the self-similar plant M1. This increase is less pronounced in the randomized plants.

The corresponding profiles for the plant family M2 are shown in Fig. 10. We observe that the paracladial coefficient γ(v) reaches a minimum for the longest lateral branches with the rank(v) = 10, 11, 12, but the weighted paracladial coefficient γ ′ (v) reaches its maximum for the same branches. The use of weights can thus qualitatively affect our assessment of the contribution of individual branches to the self-similarity of the whole plant structure.

The profiles for family M3 (Fig. 11) have a distinctly different character from the profiles of families M1 and M2, which reflects the non-self-similar character of plants in M3.

APPLICATION TO REAL PLANTS 4.1 Plant Material

We considered branching structures of two plant parts with marked self-similar organization: lilac inflorescences and rice shoots.

Five Syringa vulgaris (common lilac) inflorescences were collected and measured in Calgary Canada, in the spring of 2001. In addition to the topological structure (map) of these inflorescences, the length and the diameter of each internode, and the length and diameter of each flower were measured using a digital caliper connected to a computerized data collection system. [START_REF] Mündermann | Inverse modeling of plants[END_REF] This made it possible to reconstruct these inflorescences with great accuracy (Fig. 12). In the present study, we only used topological data. Since lilac inflorescences have two branches at each node, we averaged the paracladial coefficients of both branches to define a single value at each node on the paracladial coefficient profiles.

The second plant was an Oryza sativa (rice) cv 'Nippon Bare' plant, which has more complex lateral structures (reiterated systems [START_REF] Barthélémy | Canopy architecture[END_REF] ) than the lilac inflorescences. The topological structure of an individual plant was completely mapped, including vegetative and floral parts. [START_REF] Paul-Victor | Diversité morphologique et architecturale de quatre variétés de riz (Oryza sativa et Oryza glaberrina)[END_REF] Figure 13 shows a picture of this individual and its corresponding schematic representation. The structure is made of a main axis bearing a main inflorescence (panicle) and four lateral reiterated systems (called tillers), each composed of a vegetative part and one inflorescence. The tillers themselves bear lateral axes, some of them bearing inflorescences. This gave us the opportunity to evaluate the self-similarity of both the inflorescences and the vegetative parts.

Results

Lilac

Lilac inflorescences have high coefficients of self-similarity (average 0.92, Table 2). This MSC value is comparable to that of the randomized theoretical plants in families M1

and M2, though it is higher than even that of the least randomized plants M1 -0.8 and

M2 -0.8.
For all inflorescences, the paracladial coefficients of the first-order branches up to rank 6 (from the tip) are very close to one. The coefficients then slowly decrease to 0.6 (Fig.

14.a)

. The mean values of these coefficients range between 0.89 and 0.94 over the five inflorescences, with a low standard deviation (close to 10% on average, Table 2). This reveals a very homogeneous self-similar nature of these inflorescences, as confirmed by the almost perfectly superimposed MSC profiles (Fig. 14.b).

The weighted paracladial coefficients relate the paracladial coefficients to the complexity of branches. As shown in Fig. 14.c, the values of WPC tend to increase up to rank 9, where they stabilize around a constant value. From rank 9 on, the decrease in the value of (unweighted) paracladial coefficients is thus compensated by the fact that the structures become more complex.

To visualize these results in a more intuitive manner, we colored the branches of the reconstructed lilac inflorescences according to the values of their unweighted or weighted paracladial coefficients. If the unweighted paracladial coefficients are used (Fig. 15), the most self-similar part of each inflorescence is situated near its top, where the branches are short. Weighted paracladial coefficients compensate for this bias toward the short branches, giving a different perspective of the distribution of self-similarity within the inflorescences (Fig. 16). Contribution to self-similarity is now low at the tips, while the branches that most contribute to self-similarity are located in the medial and basal parts of the inflorescences.

In summary:

• The degree of self-similarity of the five lilac inflorescences is high (average 0.92).

• The self-similar nature of these inflorescences is very homogeneous.

• A stable trade-off between paracladial quality and complexity of lateral inflorescences is reached in the basal part of these plants (after rank 9).

• The 3D representation of paracladial coefficients gives a very intuitive visual indication of the contribution of each branch to the plant's overall self-similarity.

Rice

The topology of rice is rather complex since it contains inflorescences (panicles), vegetative parts, and reiterated systems. This gave us the opportunity to compare the self-similarity of both inflorescences and vegetative parts.

We first analyze the self-similarity of plant panicles P 1 -P 5 (Fig. 17, ranks 1-11). Their self-similarity coefficients show that the two basal panicles are highly selfsimilar (MSC equal to 0.96 and 0.97), whereas the two intermediate panicles have lower coefficients (0.90 and 0.83) (Table 3). The main stem panicle has a much lower MSC (0.69), because its axis carries long branches with relatively low paracladial coefficients, located near the tip of this panicle (Fig. The tillers themselves vary in their self-similar nature. V 2, V 4, and V 5 have similar structure for their distal part (up to rank 15) (Fig. 17.c). However, unlike V 4 and V 5,

V 2 bears a secondary tiller, which makes it more similar to the trunk and thus greatly increases its overall weighted paracladial coefficient on the basal part, to the level of V 3.

This leads V 2 and V 3 to exhibit nearly identical coefficients of self-similarity.

From this analysis, we conclude that:

• The tiller panicles are more self-similar than the apical panicle (Fig. 17.b).

• The degree of self-similarity of the tiller panicles is high (average 0.92)

• The degree of self-similarity of the whole tillers is also high (average 0.87)

• The degree of self-similarity of the entire plant is relatively lower (0.69). This is mainly due to the lack of self-similarity in the main stem panicle.

• Nevertheless, the tillers are highly similar to the main stem structure (Fig. 17.c).

This supports the idea that tillers can be viewed as reiterated complexes of the plant.

CONCLUSION

This paper addressed the problem of quantifying the degree of self-similarity in branching plant structures. To this end, we introduced the notions of paracladial coefficient and mean self-similarity coefficient for axial trees. The paracladial coefficient characterizes the similarity between an individual branch and the main stem of the structure, whereas the mean self-similarity coefficient provides a global measure of the self-similarity of the entire structure. Weighted coefficients were also introduced to take into account the size of branches while quantifying self-similarity. These definitions have been applied to both simulated branching structures and real plants. The simulated structures were used to illustrate main characteristics of the proposed measures. Real plants were used to show that these measures are appropriate for interpreting experimental data.

For over fifty years, botanists have postulated that describing a plant as an assembly of similar parts plays a key role in the understanding of plant structure and development.

The formalization and quantification of self-similarity of plant structure may contribute to this understanding. Insights from the study of self-similarity may also assist in the construction of simulation plant models, by exposing the repetitive elements of plant architecture. Furthermore, the use of self-similarity may lead to significant simplifications of plant mapping and measurement techniques, since parts known to be similar to other parts need not be measured.

The definition of self-similarity proposed in this paper is well adapted to the analysis of monopodial plants (with a clearly defined main axis or trunk). Its extensions to sympodial plants, and to alternative definitions of plant self-similarity 0.84 0.17 0.77 0.17 V 5 0.88 0.12 0.84 0.12 Table 3: Self-similarity coefficients and their standard deviation for rice panicles, the whole main axis V 1, and tillers V 2 -V 5

Figures

  is a non-negative real number less than 1. D(T 1 , T 2 ) asymptotically approaches 1 when T 1 and T 2 each have a large number of vertices and represent completely different structures (see Fig. 5.a). D(T 1 , T 2 ) = 0 if and only if

  |B(r)| = l • b numbers, where l = rank(r) is the length of the trunk and b is the branching ratio on the trunk, i.e. the mean number of lateral branches per node of the trunk. Each number γ(v) can be computed in a time proportional to the square of the size of the branching system T [v] using the algorithm described by Ferraro and Godin. 25 If m = max v∈B(r) |T [v]|, the self-similarity of the structure T [r] can be tested in time m 2 • l • b in the worst case.

7

  .b. The dashed arrows indicate state changes associated with the production of branches. The state transitions represented by these arrows relate the state s of the apical meristem with the state s ′ of the lateral meristem. Differences in these transitions are the key feature distinguishing the three families of generated branching structures, M1, M2 and M3, discussed next. The differentiation graph of each plant has seven states, with 1 denoting the initial state and 7 denoting the terminal (flowering) state. A family Mi consists of a deterministic model Mi and five derived models. The differentiation graphs of the deterministic models M1, M2 and M3 are shown in Fig. 7. In model M1, the lateral meristems that are generated by an apical meristem in state s have state s + 1. The state of the apical meristem remains unchanged for the given number of steps, then advances by 1 (except

  .a).The panicles have relatively lower paracladial coefficients in their medial parts. At the base of the panicles the paracladial coefficients of P 2, P 4 and P 5 become much higher (Fig.17

  .a). The length of the lateral branches in the panicles tends, however, to vary in the opposite way: it is high in the medial part, and lower at the base (Fig.17

  .d). The superposition of these trends results in high and relatively constant weighted paracladial coefficients for all branches up to rank 11 (Fig.17

  .c). Now let us take into account the vegetative parts of the plants in the evaluation of self-similarity. On Fig.17.a, ranks 1-21, we observe that the paracladial coefficients of all axes tend to decrease in proximal positions with respect to the panicle base, except for the main axis (V 1), for which the opposite tendency is observed. Due to the important length of tillers, this phenomenon is even more marked on the weighted paracladial coefficients (Fig.17

  .c).

Figure 1 :

 1 Figure 1: Examples of plants showing remarkably self-similar branching structures: a fern leaf, a compound inflorescence (lilac), and a romanesco broccoli.

Figure 2 :

 2 Figure 2: Different sub-structures and sets in an axial tree. a) Visualization using a schematic plant representation. b) Visualization using the equivalent axial tree representation. c) Definition of the set B(r) for an axial tree T [r].

Figure 3 :

 3 Figure 3: Perfectly self-similar branching structures with different maximum branching orders (equal to 1,2,3,4 and 1, a to e) and different apparent complexity.

Figure 4 :

 4 Figure 4: The recursive principle of self-similarity in plants. a) A visualization using a single branching structure. b) A visualization using a general representation of branching structures.

Figure 5 :

 5 Figure 5: a) Two tree structures of the same size with maximum topological distance. b) Two different tree structures that are isomorphic if axial information is not taken into account, but non-isomorphic otherwise.

Figure 6 :

 6 Figure 6: Examples of non-self-similar structures. a) The upper part is perfectly selfsimilar and the basal part is not. b) The basal part is perfectly self-similar and the upper part is not. c) An extremely non-self-similar structure. The paracladial coefficient γ(v) of each comb-like branch is equal to 2n+2 , where n is the number of components in the branch. The self-similarity coefficient of the whole plant tends to 0 as n becomes large.

Figure 7 :

 7 Figure 7: Differentiation graphs used for defining theoretical plants. a) The differentiation graph of non-branching growth. b) The resulting axis structure, where component colors correspond to the differentiation graph states in which these components were created. The numbers attached to each loop indicate the number of steps a meristem stays in the corresponding state. M1, M2, M3) The differentiation graphs of models M1, M2, and M3. Solid arrows correspond to possible transitions of the apical meristem states. Dashed arrows correspond to possible transitions from the apical meristem state to the axillary meristem states.

Figure 8 :

 8 Figure 8: Theoretical plants: the M1 family (first row), the M2 family (second row), and the M3 family (third row).

Figure 9 :

 9 Figure 9: Profiles of PC, WPC •N, MSC and α • N for theoretical plants M1. Both WPC α profiles have been multiplied by the number of branches in the plant, denoted by N.

Figure 12 :

 12 Figure12: One of the lilac inflorescences used in the analysis of self-similarity and its 3D reconstruction .[START_REF] Prusinkiewicz | The use of positional information in the modeling of plants[END_REF]

Figure 13 :

 13 Figure 13: a) Oryza sativa cv 'Nippon Bare' (rice) individual (Photo: Cloé Paul Victor). b) Schematic representation of the individual topological structure, showing one main axis and four lateral tillers. Vegetative parts are in green and inflorescences are in red. Flowers are represented as ovoid black shapes. Leaves are not represented. Labels identify the different analyzed branching structures and are located at the base of the corresponding structures.

Figure 14 :

 14 Figure 14: Profiles of PC, WPC •N, MSC and α • N for lilac inflorescences.

Figure 15 :

 15 Figure 15: False color illustrations of lilac inflorescences showing the value of the paracladial coefficient (PC).

Figure 16 :

 16 Figure 16: False color illustrations of lilac inflorescences showing the value of the weighted paracladial coefficient (WPC).

Table 1 :

 1 [START_REF] Mündermann | Inverse modeling of plants[END_REF][START_REF] Prusinkiewicz | Self-similarity in plants: Integrating mathematical and biological perspectives[END_REF] need to be further investigated. Self-similarity coefficients and their standard deviation for plant families M1,

	Tables
	M1	MSC	s.d.(MSC)	W MSC	s.d.(W MSC)
	M1 M1 -0.8 M1 -0.6 M1 -0.4 M1 -0.2 M1 -0.0	1.00 0.84 0.74 0.63 0.58 0.52	0.00 0.12 0.19 0.25 0.28 0.25	1.00 0.79 0.66 0.52 0.44 0.39	0.00 0.12 0.18 0.23 0.26 0.20
	M2 M2 M2 -0.8 M2 -0.6 M2 -0.4 M2 -0.2 M2 -0.0	1.00 0.85 0.77 0.68 0.66 0.55	0.00 0.12 0.18 0.19 0.21 0.22	1.00 0.82 0.71 0.61 0.58 0.45	0.00 0.12 0.16 0.16 0.16 0.17
	M3 M3 M3 -0.8 M3 -0.6 M3 -0.4 M3 -0.2 M3 -0.0	0.40 0.40 0.39 0.32 0.30 0.33	0.00 0.09 0.07 0.17 0.13 0.00	0.40 0.40 0.39 0.32 0.30 0.33	0.00 0.09 0.07 0.16 0.13 0.00
	M2 and M3
		MSC	s.d.(MSC)	W MSC	s.d.(W MSC)
	A1 A2 A3 A4 A5	0.89 0.94 0.89 0.93 0.92	0.13 0.08 0.12 0.10 0.09	0.82 0.82 0.83 0.88 0.86	0.10 0.06 0.09 0.07 0.07

Table 2 :

 2 Self-similarity coefficients and their standard deviation for lilac inflorescences

	Panicle	MSC	s.d.(MSC)	W MSC	s.d.(W MSC)
	P 1 P 2 P 3 P 4 P 5	0.69 0.96 0.97 0.83 0.90	0.24 0.07 0.05 0.18 0.12	0.59 0.93 0.93 0.76 0.84	0.19 0.08 0.06 0.18 0.12
	Axis / tillers
	V 1 V 2 V 3 V 4	0.69 0.88 0.89	0.20 0.16 0.18	0.65 0.81 0.79	0.14 0.20 0.23

A quotiented tree is a tree on which clusters of vertices have been defined, for which the corresponding cluster structure is also a tree.

Figure 11: Profiles of PC, WPC •N, MSC and α • N for theoretical plants M3.

ACKNOWLEDGMENT

The authors thank Lars Mündermann for collecting the data on lilac inflorescences, Yves Caraglio and Cloé Paul-Victor for kindly providing us the data on rice, and Brendan Lane for insightful comments and editorial help.

with some distal portion T 1 of T (Fig. 4). We want to show that for every vertex w of T , the branch T [w] rooted in w is also isomorphic with some distal portion of T . Proof by induction on the order n of vertex w.

1. Initial step. If n = 0, the vertex w lies on the main axis of T , and the subtree T [w] is the distal portion of T rooted in w.

2. Inductive step. Assume that the proposition holds for some n ≥ 0, and consider a vertex w of order n + 1. The branch T [w] rooted in w is included in some branch

where v is a vertex of order n. From the inductive assumption it follows that T [v] is isomorphic with some distal portion T 1 of T . The image of T [w] under this isomorphism is a (distal portion of) some first-order branch B 1 , which, according to the assumption of the "if" part of the proof, is isomorphic with some distal portion T 2 of T . By transitivity of isomorphisms, T [w] is also isomorphic with T 2 .

The "only if" part of the proof. We assume that any branch B rooted in some vertex w in T is isomorphic with some distal portion T 1 of T . In particular, any branch B 1 with the base v originating at the axis is then also isomorphic with some distal portion T 1 .


Pascal Ferraro

Christophe Godin

Przemyslaw Prusinkiewicz

A structural method for assessing self-similarity in plants

Introduction

The important role of architecture in the understanding of plants [START_REF] Frijters | Developmental descriptions of branching patterns with paracladial relationships[END_REF][START_REF] Hallé | Tropical trees and forests. An architectural analysis[END_REF][START_REF] Ross | The radiation regim and the architecture of plant stands[END_REF] generates a need for investigational tools. Generic tools have already been developed to visualize plant architecture in three dimensions [START_REF] Prusinkiewicz | The algorithmic beauty of plants[END_REF], to model the development of plant structure [START_REF] Dereffye | A functional model of tree growth and tree architecture[END_REF][START_REF] Prusinkiewicz | The algorithmic beauty of plants[END_REF], to measure plant architecture [START_REF] Sinoquet | Assessment of the three-dimensional architecture of walnut trees using digitizing[END_REF], and to analyse and quantify relations between plant components [START_REF] Godin | Measuring and analyzing plants with the amapmod software[END_REF]. This paper introduces a new tool for the identification of self-similarity within plant architectures.

In 2001, Prusinkiewicz et al. [START_REF] Prusinkiewicz | The use of positional information in the modeling of plants[END_REF] proposed a method for modeling plant branching structures that exploits their self-similarity. They showed that a lilac inflorescence can be modeled using only the measurements of the longest axis of each branch order (cf. Figure 17a in [START_REF] Prusinkiewicz | The use of positional information in the modeling of plants[END_REF]). The applicability and accuracy of this technique depend on the degree of self-similarity present in the modeled structure. But similarity and self-similarity, like symmetry, are not easily quantifiable: they are usually considered as a property that is either present or absent, rather than a property that may be present to some degree [START_REF] Rosen | Symmetry discovered[END_REF]. In this paper, we propose a procedure for quantifying the degree of self-similarity in measured or simulated plants. This procedure is based on a method for comparing branching structures introduced in [START_REF] Ferraro | A distance measure between plant architectures[END_REF].

Biological motivation

In the growth processes of many living organisms, especially plants, regularly repeated appearances of closely related multicellular structures are readily noticeable. In the simplest case, one observes the same structure being periodically repeated along an axis, such as leaves along a stem. In more complicated cases, the entire structure of a previous stage is repeated as part of the organism at a later stage, resulting in "compound organisms or organs, [. . . ], such as compound branching systems" [START_REF] Herman | Description of developmental languages using recurence system[END_REF]. This phenomenon of self-similarity [START_REF] Mandelbrot | The fractal geometry of nature[END_REF] has been tentatively captured by several botanical notions. For example, Troll defined an inflorescence as a system consisting of the main florescence and of paracladia [START_REF] Frijters | Developmental descriptions of branching patterns with paracladial relationships[END_REF]; a paracladium is "a branch which bears the same sort of structures like those borne on the terminal portion of its mother branch" [START_REF] Lindenmayer | Paracladial relationships in leaves[END_REF]. Self-similarity is also related to the definition of the physiological age of meristems in vegetative parts of plants [START_REF] Barthélémy | Levels of organization and repetition phenomena in seed plants[END_REF][START_REF] Gatsuk | Age states of plants of various growth forms: a review[END_REF]. According to this concept, all the meristems of a plant undergo the same series of developmental stages (called physiological ages), and two meristems with the same physiological age produce essentially similar structures [START_REF] Arber | Natural philosophy of plant form[END_REF][START_REF] Honda | Description of the form of trees by the parameters of tree-like body[END_REF].

These notions are reflected in several formal approaches to the description and modelling of plants. For example, Fritjers and Lindenmayer [START_REF] Frijters | Developmental descriptions of branching patterns with paracladial relationships[END_REF], defined paracladial relationships in branching patterns: a branching structure exhibits a paracladial relationship if "the behaviour of a branch is exactly the same as the behaviour of that part of the mother branch which has developed since the formation of the daughter branch". Prusinkiewicz et al. introduced the concept of branch mapping [START_REF] Prusinkiewicz | The use of positional information in the modeling of plants[END_REF], which holds if "given two branches of the same order, the shorter branch is identical [. . . ] to the top portion of the longer branch". This concept was used to reduce the complexity of three-dimensional models of branching structures, including compound inflorescences and tree structures. In the context of growth modeling, the notion of physiological age was exploited to model the development of plants by de Reffye et al. [START_REF] De Reffye | Architecture et modélisation de l'orme du japon zelkova serrata (thunb.) makino (ulmaceae): la notion d'axe de référence[END_REF], who introduced the notion of the reference axis as a list of all the differentiation stages of a tree meristem. By studying the different developmental stages of Zelkova serrata (Japanese elm), they showed that growth and branching processes are similar in any part of the tree and that it is possible to identify a theoretical axis that includes all the states of morphological differentiation of a growth unit. In this sense, the "reference axis" captures the behaviour of a typical meristem in a plant over time, i.e. the potential self-similar development of a plant.

In reality, the self-similarities captured by paracladial relationships, branch mapping or physiological age are only approximate. While easy to observe in obvious cases, they may be hidden in more complex structures [START_REF] Costes | Modelling the sylleptic branching on one-year-old trunks of apple cultivars[END_REF]. Here, we propose a computational method to quantify the similarity between different branching structures within a plant and thus quantitatively assess paracladial relationships. This method can be applied to plants represented as tree structures.

Evaluation of similarity between branching systems

A simple approach for comparing two branching systems consists of summarizing each individual by a small number of synthetic variables (eg. fruit production or crown size). The similarity between the systems is then reduced to the proximity of these synthetic variables. In contrast, our interest is in the similarity of structures. This requires a definition of distance that would take into account the topological and spatial organization of entities into branching structures. In the following, we will use the common description of plants as mathematical (graph theoretic) trees [START_REF] Godin | A multiscale model of plant topological structures[END_REF][START_REF] Prusinkiewicz | The algorithmic beauty of plants[END_REF].

A plant is considered as a connected assembly of botanical entities such as internodes or annual growth increments [START_REF] Bell | Plant form, an illustrated guide to flowering morphology[END_REF]. When describing plants at the architectural level, we assume that each such entity is physically attached to at most one parent entity. The resulting topological structure is a rooted tree graph T ¡ V¢ E £ , i.e. a graph in which every vertex except one (the root r) has only one parent vertex [START_REF] Godin | A multiscale model of plant topological structures[END_REF]. The root has no parent vertex. In the following a tree graph rooted in r will be represented by T ¤ r¥ , and the empty tree graph will be denoted by θ ¡ / 0¢ / 0£ . In order to identify the different axes on a given plant, two types of relations between entities are distinguished: an entity can either precede (symbol '¦ ') or bear (symbol ' § ') another entity. We suppose that an entity can be attached to only one other entity by a connection '¦ '. A sequence of entities connected by '¦ ' forms an axis of a tree [START_REF] Prusinkiewicz | The algorithmic beauty of plants[END_REF].

The evaluation of similarity between branching structures has been largely studied in computer science and is known as the tree-to-tree comparison problem [START_REF] Ohmori | A unified view on tree metrics[END_REF]. The distance between two tree graphs is equal to the minimum cost of the sequence of edit operations needed to transform one tree graph into the other. We consider three kinds of elementary edit operations on a tree graph T [START_REF] Zhang | A constrained edit distance between unordered labeled trees[END_REF]: substituing one vertex for another (note that this changes their labels), deleting a vertex, and inserting a vertex. A constraint is added to the definition of insertions and deletions: we only consider insertions of leaves or vertices between a parent vertex and its set of children; deletions are similarly constrained 1 This dissimilarity measure coincides with the definition of a distance between unordered 2 tree graphs proposed by Zhang [START_REF] Zhang | A constrained edit distance between unordered labeled trees[END_REF]. He proposed a polynomial algorithm to recursively calculate the cost of the optimal sequence of edit operations, which was applied to compare plant architectures by Ferraro and Godin [START_REF] Ferraro | A distance measure between plant architectures[END_REF]. In that case, the computed distance strongly depended on the size of the compared tree graphs. In order to make the comparison results less dependent on this size, we normalize them by dividing the distance by the total number of vertices in compared tree graphs: D¡

T 1 ¢ T 2 £ D ¡ T 1 ¢ T 2 £ ¡ ¡ T 1 § T 2 £
This dissimilarity measure is a non-negative real number less than 1.

Quantification of tree self-similarity

In this section we extend the previous algorithm to identify and quantify self-similarity of plant structures.

Let us consider a plant P represented by a tree graph T ¤ r¥ . The set of all sub-tree graphs of T ¤ r¥ is denoted by T ¡ r£ . Let v be a vertex of T ¤ r¥ and let T ¤ v¥ be a sub-tree graph rooted in v of T ¤ r¥ ; then the axis A¤ v¥

A[v] v A ( v ) v A[r]
¡ V¢ E £ of T ¤ v¥ is the sub-graph of T ¤ v¥ such that V £¢ v 1 ¢ v 2 ¢ ¢ v n ¤ , v 1 v, and for any i in ¢ 1 n ¥ 1 ¤ , ¡ v i ¢ v i¦ 1 £ is
an edge of type '¦ ' (Figure 1). A ¡ r£ is called the main axis of P (i.e., the trunk).

If T ¤ v¥ is a sub-tree of P and A¤ v¥ is the axis of T ¤ v¥ starting at v, we denote by A ¡ v£ the set of sub-trees of P rooted in a vertex of A¤ v¥ :

A ¡ v£ £¢ T ¤ v 1 ¥ ¢ T ¤ v 2 ¥ ¢ T ¤ v 3 ¥ ¢ ¢ T ¤ v n ¥ ¤ Note that, in this definition, v i¦ 1 is necessarily a child of v i . A
¡ v£ is thus a set of embedded tree graphs(Figure 1). Using the above notions, we quantify the similarity of a branch to the main axis as follows:

Definition 1 For any sub-tree graph T ¤ v¥ in T ¡ r£ , we call the paracladial coefficient of T ¤ v¥ the quantity pr ¡ v£ min

T § r i¨ A © r D¡ T ¤ v¥ ¢ T ¤ r i ¥ £
2 An unordered tree graph is one in which there is no ordering distinction among the children of any vertex.

The paracladial coefficient is thus a measure of similarity between a given branch and the top portion of the main axis that is the closest (in the sense of distance D) to that branch. In order to evaluate self-similarity of an entire structure, we compute statistical parameters such as the mean value of the distance and its variance:

PR 1 T ¡ r£ ¡ ∑ v T pr ¡ v£ ¢ var ¡ PR£ 1 T ¡ r£ ∑ v T ¡ pr ¡ v£ ¥ PR£ 2
Here T ¡ r£ represents the number of sub-tree graphs in T ¡ r£ . Low values of PR indicate high selfsimilarity of the described branching structure.

Example

In this section we apply our method to assess The length and the diameter of each internode, the spatial orientation of each branch, and the length and the diameter of each flower have been measured using a digital calliper connected to a computerized data collection system. The data were analyzed using AMAPmod [START_REF] Godin | Measuring and analyzing plants with the amapmod software[END_REF], a software dedicated to plant architecture analysis, in which the algorithms of comparison between tree-graph structures have been implemented. The results were visualized using the modeling program cpfg [START_REF] Měch | CPFG version 3.4 user's manual[END_REF]. ¡ v¢ w£ of vertices (botanical internodes). These definitions capture topological self-similarity [START_REF] Prusinkiewicz | Self-similarity in plants: Integrating mathematical and biological perspectives[END_REF] of the architecture (i.e., the topological organization of entities in the branching system), without taking into account any geometrical attributes. The paracladial coefficients pr ¡ v£ of all the order-2 branches are visualized in Figure 2. Generally, these coefficients are very small: the inflorescences show only a few "errors" in the mapping. This high degree of self-similarity is reflected in the statistics collected in Table 1.

Conclusion

This paper addressed the problem of quantifying the degree of self-similarity in branching plant structures. To this end, we introduced the notion of paracladial index and its statistics. The index characterizes the similarity between an individual branch and the main stem of the structure, whereas the statistics provide a global measure of self-similarity of the entire structure.

The presented results can be applied to improve the methodologies of measurement and architectural modeling of plants. For example, when measuring plants, it may be unnecessary to consider all branches, because of their likely similarity to the main branch. When modeling plants, some of the data characterizing the main branch may be re-used in the lateral branches, thus simplifying the models. The paracladial index and its statistics make it possible to quantify the resulting errors.

Fig. 1 .

 1 Fig. 1. Axis definition in a branching system (a) and in its representation as a tree graph (b).

Fig. 2 .

 2 Fig. 2. Color-coded representation of paracladial coefficients of branches of order 2 self-similarity in a sample of five Syringa vulgaris (common lilac) inflorescences. The inflorescences were collected in Calgary in the spring of 2001.The length and the diameter of each internode, the spatial orientation of each branch, and the length and the diameter of each flower have been measured using a digital calliper connected to a computerized data collection system. The data were analyzed using AMAPmod[START_REF] Godin | Measuring and analyzing plants with the amapmod software[END_REF], a software dedicated to plant architecture analysis, in which the algorithms of comparison between tree-graph structures have been implemented. The results were visualized using the modeling program cpfg[START_REF] Měch | CPFG version 3.4 user's manual[END_REF].To evaluate self-similarity, we first need to PR var

¡

  v£ for five lilac inflorescences, and the percentage of branches of order 2 perfectly similar to top portions of their supporting branches. define a local distance between elementary entities of the branching systems. In our example, we used a topological distance, with the elementary distance d defined as d ¡ v¢ w£ 0 and the distance d indel defined as d indel ¡ v£ 1 for any pair

  . A cost function is defined for each edit operation s which assigns a non-negative real number γ Here d is a distance called the elementary distance, defined on the set of vertices of T . We assume, for any pair of vertices v and w, that d ¢ s 2 ¢ ¢ s n £ which transforms a tree graph T 1 into another tree graph T 2 . The cost Γ

	¡ s£ to s as follows: γ	¡ s£	d	¡ v¢ w£ if s is a substitution of v by w; and γ	¡ s£	d indel	¡ v£ if s is an
	insertion or a deletion of vertex v. ¡ v¢ w£ ©¨d indel Let S be a sequence of n edit operations ¡ s 1	¡ v£ § d indel	¡ w£ .

¡

S£ of S is defined by summing up the cost of the edit operations that compose S:

Γ ¡ S£ ∑ s i S γ ¡ s i £ . The dissimilarity measure D ¡ T 1 ¢ T 2

£ between a tree graph T 1 and a tree graph T 2 is then defined as the minimum cost of a sequence in S that transforms T 1 into T 2 .

Table 1 .

 1 To evaluate self-similarity, we first need to PR var The mean value PR and variance var

			¡ PR£	nb
	A 1 A 2 A 3 A 4 A 5	0 0164 0 0196 0 0212 0 0227 0 0192	0 00298 0 00247 0 00315 0 00223 0 00336	70 83% 78 26% 68 18% 79 16% 72 72%

¡

PR£ of the paracladial index pr

Zhang and Jiang[START_REF] Zhang | Some max snp-hard results concerning unordered labeled trees[END_REF] have shown that the computation of distance between tree graphs using such edit operations is a MAX SNPhard problem unless constraints are added to the definition of edit operations.

Acknowledgment

The authors thank Lars Mündermann for sharing with us his database of lilac inflorescence measurements, and for contributing to the model of these inflorescences.