Philippe Jacquet

Svante Janson

VARIATIONS ON DEPTH-FIRST SEARCH PERFORMANCE IN RANDOM DIGRAPHS

Keywords: Combinatorics, Depth-First Search, Random Digraphs

We present an analysis of the depth-first search algorithm in a random digraph model with geometric outdegree distribution. This problem posed by Don Knuth in his next to appear volume of The Art of Computer Programming gives interesting insight in one of the most elegant and efficient algorithm for graph analysis due to Tarjan.

Introduction

The motivation of this paper is a new section in Donald Knuth's The Art of Computer Programming [START_REF] Knuth | The Art of Computer Programming[END_REF], which is dedicated to Depth-First Search (DFS) in a digraph. We refer to [START_REF] Knuth | The Art of Computer Programming[END_REF] for the definition of DFS as well as for historical notes. Note that the digraphs in [START_REF] Knuth | The Art of Computer Programming[END_REF] and here are multi-digraphs, where loops and multiple arcs are allowed. The DFS algorithm generates a spanning forest (the depth-first forest) in the digraph, with all arcs in the forest directed away from the roots. Our main purpose is to study the distribution of the depth of vertices in the depth-first forest, starting with a random digraph G.

Furthermore, the DFS algorithm in [START_REF] Knuth | The Art of Computer Programming[END_REF] classifies the arcs in the digraph into the following five types, see Figure 1 for examples:

• loops; • tree arcs, the arcs in the resulting depth-first forest;

• back arcs, the arcs which point to an ancestor of the current vertex in the current tree; • forward arcs, the arcs which point to an already discovered descendant of the current vertex in the current tree; • cross arcs, all other arcs (these point to an already discovered vertex which is neither a descendant nor an ancestor of the current vertex, and might be in another tree).

We will discuss the numbers of arcs of different types. (See further the exercises in [START_REF] Knuth | The Art of Computer Programming[END_REF].)

The random digraph model that we consider has n vertices and a given outdegree distribution P. The outdegrees (number of outgoing arcs) of the n vertices are independent random numbers with this distribution. The endpoint of each arc is uniformly selected at random among the n vertices, independently of all other arcs. (Therefore, an arc can loop back to the starting vertex, and multiple arcs can occur.) We consider asymptotics as n → ∞ for a fixed outdegree distribution.

We will focus on the case of a geometric outdegree distribution; the lack-of-memory property in this case leads to interesting features and a simpler analysis. The paper will study the following outdegree distribution in the following order:

• a geometric distribution; • a shifted geometric distribution (starting from integer 1 instead of zero); • a general distribution. Example of a depth-first forest (jungle) from [START_REF] Knuth | The Art of Computer Programming[END_REF], by courtesy of Donald Knuth. Tree arcs are solid (e.g. 9 ○→ 3 ○). For example, 3 The mean outdegree, i.e., the expectation of P, is denoted by λ.

○ 3 ○ is a loop, 2 ○ 3 ○ is a back arc, 9 ○ 7 ○ is a forward arc, 8 ○ 4 ○

Depth analysis with geometric outdegree distribution

Sgeo

In this section we assume that the outdegree distribution is geometric Ge(1 -p) for some fixed 0 < p < 1, and thus has mean λ := p/(1 -p). Note that in the DFS, when we find a new vertex v, we do not have to immediately reveal its outdegree. Instead, we only check whether there is at least one outgoing arc (probability p), and if so, we find its endpoint and explores this endpoint if it has not already been visited; eventually, we return to v, and then we check whether there is another outgoing arc (again probability p, by the lack-of-memory property of the geometric distribution), and so on.

In the following, by a future arc from some vertex, we mean an arc that at the current time has not yet been seen by the DFS.

2.1. Depth Markov chain. Our aim is to track the evolution of the search depth as a function of the number of t of discovered vertices. Let v t be the t-th vertex discovered by the DFS (t = 1, . . . , n), and let d(t) be the depth of v t in the resulting depth-first forest, i.e., the number of tree edges that connect the root of the current tree to v t . The first found vertex v 1 is a root, and thus d(1) = 0.

The quantity d(t) follows a Markov chain with transitions (1 ≤ t < n):

(i) d(t + 1) = d(t) + 1.
This happens if, for some k ≥ 1, v t has at least k outgoing arcs, the first k -1 arcs lead to vertices already visited, and the kth arc leads to a new vertex (which then becomes v t+1 ). The probability of this is

∞ k=1 p k t n k-1 1 - t n = (1 -t/n)p 1 -pt/n . (1) new (ii) d(t + 1) = d(t), assuming d(t) > 0.
This holds if all arcs from v t lead to already visited vertices, i.e., (i) does not happen, and furthermore, the parent of v t has at least one further arc leading to an unvisited vertex. These two events are independent. Moreover, by the lack-of-memory property, the number of further arcs from the parent of v t has the same distribution Ge(1 -p).

Hence, the probability that one of these further arcs leads to an unvisited vertex equals the probability in [START_REF] Knuth | The Art of Computer Programming[END_REF]. The probability of (ii) is thus

1 - (1 -t/n)p 1 -pt/n (1 -t/n)p 1 -pt/n . (2) ii (iii) d(t + 1) = d(t) -ℓ, assuming d(t) > ℓ ≥ 1.
This happens if all arcs from v t lead to already visited vertices, and so do all further arcs from the ℓ closest ancestors of v t , while the (ℓ + 1)th ancestor has at least one further arc leading to an unvisited vertex. The argument in (ii) generalizes and shows that this has probability

1 - (1 -t/n)p 1 -pt/n ℓ+1 (1 -t/n)p 1 -pt/n . (3) iii (iv) d(t + 1) = d(t) -ℓ, assuming d(t) = ℓ ≥ 0.
By the same argument as in (ii) and (iii), except that the (ℓ + 1)th ancestor does not exist and we ignore it, we obtain the probability

1 - (1 -t/n)p 1 -pt/n ℓ+1 .
(4) iv Note that (iv) is the case when d(t + 1) = 0 and thus v t+1 is the root of a new tree in the depth-first forest. We can summarize (i)-(iv) in the formula

d(t + 1) = d(t) + 1 -ξ t + , (5) dt+
where x + := max{x, 0}, and ξ t is a random variable, independent of the history, with the distribution

P(ξ t = k) = (1 -π t ) k π t , k ≥ 0, with π t := (1 -t/n)p 1 -pt/n = 1 - 1 -p 1 -pt/n . ( 6
) xi
In other words, ξ t has the geometric distribution Ge(π t ). Define

d(t) := t-1 i=1 (1 -ξ i ), (7) td
and note that (7) is a sum of independent random variables. Then (5) and induction yield

d(t) = d(t) -min 1≤j≤t d(j), 1 ≤ t ≤ n. (8) dtd
We can also express these relations using generating functions. Let p(t, z) be the probability generating function E z ξt of ξ t , i.e., t) ]. We then have the identity, equivalent to (5),

p(t, z) := (1 -t/n)p 1 -pt/n ℓ≥0 1 - (1 -t/n)p 1 -pt/n ℓ z ℓ = (1 -t/n)p 1 -pt/n -(1 -p)z , ( 9
) ptz and let f (t, z) := E[z d(
f (t + 1, z) = N R(t, z)f (t, z) (10)
where R(t, z) := p(t, 1/z)z and N is the operator on power series in z ±1 :

N g(z) = Π + g(z) + Π -g(1) (11)
where Π + is the operator which removes the strictly negative powers of z and Π -is the operator which removes the non-negative powers of z. Thus we have, since f (1, z) = 1,

f (t + 1, z) = N R(t, z)N R(t -1, z)N • • • N R(1, z). (12)
2.2. Main result for depth analysis. Note first that (7) implies that the expectation of

d(t) is E d(t) = t-1 i=1 (1 -E ξ i ) = t-1 i=1 1 - 1 -π i π i = t-1 i=1 1 - 1 -p p(1 -i/n) . ( 13
) Etd1
Let θ := t/n. We fix a θ * < 1 and obtain that, uniformly for θ ≤ θ * ,

E d(t) = t 0 1 - 1 -p p(1 -x/n) dx + O(1) = n ℓ(θ) + O(1), (14) Etd
where

ℓ(θ) := θ 0 1 - 1 -p p(1 -τ ) dτ = θ + 1 -p p log(1 -θ). ( 15
) tl Note that the derivative ℓ ′ (θ) = 1 -(1 -p)/(p(1 -θ)) is (strictly) decreasing on (0, 1), i.e., ℓ is concave. Moreover, if p > 1 2
, which we call the supercritical case, then ℓ ′ (0) > 0, and (15) shows that ℓ(θ) is positive and increasing for θ < θ 0 := (2p -1)/p. After the maximum at θ 0 , ℓ(θ) decreases and tends to -∞ as θ ↗ 1. Hence, there exists a 0 < θ 1 < 1 such that ℓ(θ 1 ) = 0; we then have ℓ(θ) > 0 for 0 < θ < θ 1 and ℓ(θ) < 0 for θ > θ 1 . We will see that in this case the depth-first forest w.h.p. contains a giant tree, of order and height both linear in n, while all other trees are small.

On the other hand, if p ≤ 1 2 (the subcritical and critical cases), then ℓ ′ (0) ≤ 0 and ℓ(θ) is negative and decreasing for all θ ∈ (0, 1). In this case, we define θ 0 := θ 1 := 0 and note that the properties just stated for ℓ still hold (rather trivially). We will see that in this case w.h.p. all trees in the depth-first forest are small.

Note that in all cases, θ 1 is the largest solution in [0, 1) to, recalling λ = p/(1 -p),

log(1 -θ 1 ) = -λθ 1 .
(16) gth1

Rgth1

Remark 2.1. The equation ( 16) may also be written 1-θ 1 = exp(-λθ 1 ), which shows that θ 1 is the survival probability of a Galton-Watson process with Po(λ) offspring distribution.

We define ℓ + (θ) := [ ℓ(θ)] + . Thus, by (15) and the comments above,

ℓ + (θ) = θ + 1-p p log(1 -θ), 0 ≤ θ ≤ θ 1 , 0, θ 1 ≤ θ ≤ 1. ( 17
) tlp
We can now state one of our main results. Proofs are given in the next subsection.

T1

Theorem 2.2. We have

max 1≤t≤n d(t) -nℓ + (t/n) = O L 2 (n 1/2 ). ( 18
) t1 CH Corollary 2.3. The height Υ of the depth-first forest is Υ := max 1≤t≤n d(t) = υn + O L 2 (n 1/2 ), ( 19
) gU where υ = υ(p) := ℓ + (θ 0 ) = 0, 0 < p ≤ 1/2, 2p-1 p -1-p p log p 1-p , 1/2 < p < 1.
(20) gu Moreover, we can show that the height Υ is asymptotically normally distributed. Details will be given in the full paper.

CA

Corollary 2.4. The average depth d in the depth-first forest is

d := 1 n n t=1 d(t) = αn + O L 2 (n 1/2 ), ( 21
) ca where α = α(p) := 1 2 θ 2 1 - 1 -p p (1 -θ 1 ) log(1 -θ 1 ) + θ 1 = 2p -1 p θ 1 - 1 2 θ 2 1 (22) ga 2.3. Proofs.
Proof of Theorem 2.2. Since ( 7) is a sum of independent random variables, d(t) -E d(t) (t = 1, . . . , n) is a martingale, and Doob's inequality yields, for all T ≤ n,

E max t≤T | d(t) -E d(t)| 2 ≤ 4 E | d(T ) -E d(T )| 2 = 4 T -1 i=1 Var(ξ i ). ( 23
) emma
As above, fix θ * < 1, and assume, as we may, that θ * > θ 1 . Let T * := ⌊nθ * ⌋, and consider first t ≤ T * . For i < T * , we have Var ξ i = O(1), and thus, for

T = T * , the sum in (23) is O(T * ) = O(n). Consequently, (23) yields max t≤T * d(t) -E d(t) = O L 2 (n 1/2 ). (24)
Hence, by ( 14), Moreover, for t/n ≤ θ 1 , we have min 1≤j≤t ℓ(j/n) = O(1/n), while for t/n ≥ θ 1 , we have

M * := max t≤T * d(t) -n ℓ(t/n) = O L 2 (n 1/2 ). ( 25
min 1≤j≤t ℓ(j/n) = ℓ(t/n). Hence, for all t ≤ T * , min 1≤j≤t ℓ(j/n) = ℓ(t/n) -ℓ + (t/n) + O(1/n), (27)
and thus, by (26),

min 1≤j≤t d(j) -n ℓ(t/n) + n ℓ + (t/n) ≤ M * + O(1/n). ( 28
) ew
Finally, by (8), ( 25) and (28),

d(t) -n ℓ + (t/n) ≤ 2M * + O(1/n). ( 29
) jb
This holds uniformly for t ≤ T * , and thus, by (25),

max 1≤t≤T * d(t) -nℓ + (t/n) = O L 2 (n 1/2 ). ( 30
) jesp
It remains to consider T * < t ≤ n. Then the argument above does not quite work, because π t ↘ 0 and thus Var ξ t ↗ ∞ as t ↗ n. We therefore modify ξ t . We define π t := max{π t , π T * }; thus π t = π t for t ≤ T * and π t > π t for t > T * . We may thus define independent random variables ξ t such that ξ t ∼ Ge( π t ) and ξ t ≤ ξ t for all t < n. (Thus, ξ t = ξ t for t ≤ T * .)

In analogy with (7)-(8), we further define

d(t) := t-1 i=1 1 -ξ i , ( 31
) htd d(t) := d(t) -min 1≤j≤t d(j) = max 1≤j≤t t-1 i=j 1 -ξ i . ( 32
) hd Since ξ i ≤ ξ i , (32) implies that d(t) ≥ d(t) for all t.
We have Var ξ t = O(1), uniformly for all t < n, and thus the argument above yields

max 1≤t≤n d(t) -n[ ℓ(t/n)] + = O L 2 (n 1/2 ), (33) kasp
where

ℓ(θ) := θ 0 min 1 - 1 -p p(1 -τ ) , 1 - 1 -p p(1 -θ * ) dτ. ( 34
) htl
We have ℓ(θ) = ℓ(θ) for θ ≤ θ * , and for θ ≥ θ * , ℓ(θ) is negative and decreasing (since

θ * > θ 1 ). Hence, [ ℓ(θ)] + = ℓ + (θ) for all 0 < θ ≤ 1.
In particular, [ ℓ(θ)] + = ℓ + (θ) = 0 for all θ ≥ θ * , and (33) implies max

T * <t≤n d(t) = O L 2 (n 1/2 ). (35)
Recalling d(t) ≤ d(t), we thus have max

T * <t≤n d(t) -n ℓ + (t/n) = max T * <t≤n d(t) ≤ max T * <t≤n d(t) = O L 2 (n 1/2 ), (36)
which completes the proof. □

Proof of Corollary 2.3. Immediate from Theorem 2.2 and (15), since we have max

t ℓ + (t/n) = max θ ℓ + (θ) + O(1/n) and max θ ℓ + (θ) = ℓ + (θ 0 ) = ℓ(θ 0 ). □ Proof of Corollary 2.4. By Theorem 2.2, 1 n n t=1 d(t) = n t=1 ℓ + (t/n) + O L 2 n 1/2 = nα + O L 2 n 1/2 , (37) bal1
where

α := 1 0 ℓ + (τ ) dτ = θ1 0 ℓ(τ ) dτ = θ1 0 τ + 1 -p p log(1 -τ ) dτ = 1 2 θ 2 1 - 1 -p p (1 -θ 1 ) log(1 -θ 1 ) + θ 1 , (38) bal2
which yields (22), using (16). □ 2.4. The trees in the forest.

TT

Theorem 2.5. Let N be the number of trees in the depth-first forest. Then

N = ρn + O L 2 (n 1/2 ), ( 39
) tt where ρ = ρ(p) := 1 -θ 1 - p 2(1 -p) (1 -θ 1 ) 2 . ( 40
) rho
Proof. Let I t := 1{d(t) = 0}, the indicator that vertex t is a root and thus starts a new tree. Thus N = n 1 I t . If θ 1 > 0 (i.e., p > 1 2 ), then Theorem 2.2 shows that w.h.p. d(t) > 0 in the interval (1, nθ 1 ), except possibly close to the endpoints. Thus the DFS will find one giant tree of order ≈ θ 1 n, possibly preceded by a few small trees, and, as we will see later, followed by many small trees. To obtain a precise estimate, we note that there exists a constant c > 0 such that ℓ(θ) ≥ min{cθ, c(θ 1 -θ)} for θ ∈ [0, θ 1 ]. Hence, if t ≤ nθ 1 and d(t) = 0, then d(t) ≤ d(t) = 0 by (8) and, recalling (25),

M * ≥ n ℓ(t/n) ≥ c min{t, nθ 1 -t}. (41)
Consequently,

d(t) = 0 with t ≤ nθ 1 implies t ∈ [1, c -1 M * ] ∪ [nθ 1 -c -1 M * , nθ 1 ]. The number of such t is thus O(M * + 1) = O L 2 (n 1/2
), using (25).

Let T 1 := ⌈nθ 1 ⌉. We have just shown that (the case θ 1 = 0 is trivial)

T1-1 t=1 I t = O L 2 (n 1/2 ). ( 42
) mma
It remains to consider t ≥ T 1 . Let

µ t := E ξ t = 1 -π t π t = 1 -p p(1 -t/n) . ( 43
) mut
For any integer k ≥ 0, the conditional distribution of ξ t -k given ξ t ≥ k equals the distribution of ξ t . Hence,

E (ξ t -k) + = E ξ t -k | ξ t ≥ k P(ξ t ≥ k) = µ t P(ξ t -k ≥ 0). ( 44
) erika
We use again the stochastic recursion (5). Let F t be the σ-field generated by ξ 1 , . . . , ξ t-1 . Then d(t) is F t -measurable, while ξ t is independent of F t . Hence, (5) and ( 44) yield

E d(t + 1) | F t = E d(t) + 1 -ξ t | F t + E (ξ t -1 -d(t)) + | F t = d(t) + 1 -µ t + µ t P ξ t -1 -d(t) ≥ 0 | F t = d(t) + 1 -µ t + µ t P d(t + 1) = 0 | F t = d(t) + 1 -µ t + µ t E I t+1 | F t . ( 45
) ele
We write ∆d(t) := d(t + 1) -d(t) and I t := 1 -I t . Then (45) yields

E ∆d(t) -1 + µ t I t+1 | F t = 0. ( 46
) win Define M t := t-1 i=1 µ -1 i ∆d(i) -1 + µ i I i+1 = t-1 i=1 µ -1 i ∆d(i) -µ -1 i + I i+1 . (47) cM
Then M t is F t -measurable, and (46) shows that M t is a martingale. We have, with ∆M t := M t+1 -M t , using (5),

|∆M t | ≤ µ -1 t d(t + 1) -d(t) -1 + I t+1 ≤ µ -1 t ξ t + 1, ( 48
) cm1
and thus, since π t ≤ p < 1 for all t by (6),

E|∆M t | 2 ≤ 2µ -2 t E ξ 2 t + 2 = 2 π t 1 -π t 2 1 -π t + (1 -π t ) 2 π 2 t + 2 = O(1). ( 49
) cm2
Hence, uniformly for all T ≤ n,

E M 2 T = T -1 t=1 E |∆M t | 2 = O(T ) = O(n). ( 50
) cm3
The definition (47) yields

M n -M T1 = n-1 t=T1 µ -1 t ∆d(t) - n-1 t=T1 µ -1 t + n-1 t=T1 I t+1 . (51) cm4
By a summation by parts, and interpreting µ -1 n := 0,

n-1 t=T1 µ -1 t ∆d(t) = n t=T1+1 µ -1 t-1 -µ -1 t d(t) -µ -1 T1 d(T 1 ). ( 52
) cm5
As t increases, µ t increases by ( 43), and thus µ -1 t-1 -µ -1 t > 0. Hence, (52) implies

n-1 t=T1 µ -1 t ∆d(t) ≤ n t=T1+1 µ -1 t-1 -µ -1 t sup i>T1 |d(t)| + µ -1 T1 |d(T 1 )| ≤ 2µ -1 T1 sup i≥T1 |d(t)| = O L 2 (n 1/2 ) (53) cm6
by (18), since ℓ + (t/n) = 0 for t ≥ T 1 ≥ nθ 1 . Furthermore, (50) shows that M n , M T1 = O L 2 (n 1/2 ). Hence, (51) yields

n t=T1+1 I t = n -T 1 - n t=T1+1 I t = n -T 1 - n-1 t=T1 µ -1 t + O L 2 (n 1/2 ) = nρ + O L 2 (n 1/2 ), (54) cm7
where, recalling (43),

ρ := 1 -θ 1 - 1 θ1 p(1 -τ ) 1 -p dτ = 1 -θ 1 - p 2(1 -p) (1 -θ 1 ) 2 . ( 55
) rho=
The result follows by ( 54) and ( 42). □

The argument in the proof of Theorem 2.5 shows also the following; we omit the details.

TT2

Theorem 2.6. If p > 1 2 , then the largest tree in the depth-first forest has order θ 1 n + O L 2 (n 1/2 ).

Rslow

Remark 2.7. When p > 1 2 , the height is thus linear in n, unlike many other types of random trees. This might imply a rather slow performance of algorithms that operate on the depth-first forest.

Conjecture 2.8. If p = 1 2 , then the largest tree has order roughly n 2/3 . If p < 1 2 , then the largest tree has order roughly log n.

Types of arcs.

Recall from the introduction the classification of the arcs in the digraph G. Since we assume that the outdegrees are Ge(1 -p) and independent, the total number of arcs, M say, has a negative binomial distribution with mean λn, and, by a weak version of the law of large numbers,

M = λn + O L 2 (n 1/2 ). ( 56
) sw
In the following theorem, we give the asymptotics of the expected number of arcs of each type. We conjecture that Var B, Var F, Var C = O(n), so that (59)-( 60) can be improved to asymptotics for the random numbers similar to (58).

Tarcs

Theorem 2.9. Let L, T , B, F and C be the numbers of loops, tree arcs, back arcs, forward arcs, and cross arcs in the random digraph G.

L = O L 2 (1), ( 57
) tal T = χn + O L 2 (n 1/2 ), ( 58
) tat E F = E B = λ E d = βn + O(n 1/2 ), ( 59
) taf E C = χn + O(n 1/2 ), ( 60
)
tac
where

β := λα = 2p -1 1 -p θ 1 - λ 2 θ 2 1 , ( 61
) gb χ := 1 -ρ = θ 1 + λ 2 (1 -θ 1 ) 2 . ( 62
) chi
Remark 2.10. Part of the theorem is the exact equality E B = E F for any n and p. Knuth [START_REF] Knuth | The Art of Computer Programming[END_REF] conjectures, based on exact formulas for small n, that, much more strongly, B and F have the same distribution for every n. Moreover, Knuth [START_REF] Knuth | The Art of Computer Programming[END_REF] conjectures that E C = E T for every n. (The theorem above shows only that this holds asymptotically as n → ∞.)

Proof. L: A simple argument with generating functions shows that the number of loops at a given vertex v is Ge(1 -p/(n -np + p)); these numbers are independent, and thus [START_REF] Knuth | The Art of Computer Programming[END_REF]. Moreover, it is easily seen that asymptotically, L has a Poisson distribution, L d -→ Po(λ) T : This follows immediately from Theorem 2.5, since T = n -N . B, F : Let v, w be two distinct vertices. If the DFS finds w as a descendant of v, then there will later be Ge(1 -p) arcs from w, and each has probability 1/n of being a back arc to v. Similarly, there will be Ge(1 -p) future arcs from v, and each has probability 1/n of being a forward arc to w. Hence, if I vw is the indicator that w is a descendant of v, and B vw [F vw ] is the number of back [forward] arcs vw, then

L ∼ NegBin n, 1 -p/(n -np + p) with E L = p/(1 -p) = O(1) and Var(L) = p(1 -p + p/n)/(1 -p) 2 = O(1)
E B wv = E F vw = λ n E I vw . (63)
Summing over all pairs of distinct v and w, we obtain

E B = E F = λ n E w v̸ =w I vw = λ n E w d(w), (64) and

Depth, trees and arc analysis in the shifted geometric outdegree distribution

In this section, the outdegree distribution is Ge 1 (1 -p) = 1 + Ge(1 -p). Thus its mean λ = 1/(1 -p). As in Section 2, the depth d(t) is a Markov chain given by ( 5), but the distribution of ξ t is now different. The probability (1) is replaced by (1 -t/n)/(1 -pt/n), but the number of future arcs from an ancestor is still Ge(1 -p), and, with θ := t/n,

P ξ t = k = π t := 1-θ 1-pθ , k = 0, (1 -π t )(1 -π t ) k-1 π t , k ≥ 1, (65) xi1
where π t = pπ t is as in (6). The probability generating function of ξ t is, instead of (9),

p(t, z) = π t + (1 -π t ) π t z 1 -(1 -π t )z = (1 -θ) 1 -(1 -p)z 1 -pθ -(1 -p)z . (66)
We now have E ξ i = (1-p)θ p(1-θ) and instead of (14) we have E d(t) = n ℓ(θ) + O(1) where now ℓ(θ) takes the new value

ℓ(θ) := 1 p θ + 1 -p p log(1 -θ) (67) tl1
The rest of the analysis does not change but we get different values for the constants. Note that ℓ(θ 1 ) = 0 still gives (16), now with λ = 1/(1 -p), and that λ > 1 for every p. Differentiating (67) shows that the maximum point θ 0 = p > 0.

The results in Theorems 2.2-2.6 thus hold, with, by straightforward calculations,

υ := ℓ(p) = 1 + 1 -p p log(1 -p), (68)
α := 1 p θ 2 1 2 - 1 λ ((1 -θ 1 ) log(1 -θ 1 ) - 1 λ θ 1 = θ 1 - θ 2 1 2p , (69)
ρ := 1 -θ 1 - 1 2(1 -p) (1 -θ 1 ) 2 . ( 70
)
Figure 2 shows ℓ(θ) for both geometric distributions.

Figure 2. ℓ(θ), the asymptotic search depth, for geometric distribution (solid) and shifted geometric distribution (dashed) with p = 0.6.

fig:depth

Now the expected numbers of back and forward arcs differ since E B = λ E d ∼ λαn and E F = (λ -1) E d ∼ (λ -1)αn because the average number of future arcs at a vertex after a descendant have been created is λ -1. Thus the equality of the number of backward and forward arcs in Theorem 2.9 was an artefact of the geometric degree distribution.

The number of tree arcs is still

T = χn + O L 2 (n 1/2 ) (71) with χ = 1 -ρ = θ 1 + λ 2 (1 -θ 1 ) 2 .
But the number of cross arcs differs. We get

E C = χ ′ n + O(n 1/2 ) ( 72
)
with

χ ′ := λ -χ -λα -(λ -1)α = λ 2 1 -2θ 1 + 1 p θ 2 1 = λ 2 (1 -θ 1 ) 2 + 1 2p θ 2 1 . (73)
Again the (asymptotic) identity of E T and E C was an effect of the geometric distribution.

Stack index analysis and forest size for a general outdegree distribution

In this section, we consider a general outdegree distribution, with mean λ and finite variance. When the outdegree distribution is general, the depth does not longer follow an easy Markov chain, since we should keep track of the number of children seen so far at each level of the branch of the tree toward the current vertex.

We denote m the degree of a random vertex. When the distribution of m is general, the depth does not longer follow an easy Markov chain, since we should keep track of the number of children seen so far at each level of the branch of the tree toward the current vertex. However we can set the following theorem.

Theorem 4.1. In the general outdegree distribution when λ > 1 we have the following expansion for small values of t:

E[d(t)] = (1 -T (1))t + O(t 2 /n) ( 74
)
with T (z) the p.g.f. of the finite tree of the Galton Watson tree generated by the outdegree m.

Remark 4.2. It turns out that the theorem also holds for λ < 1, but in this case T (1) > 1.

Proof. The p.g.f. T (z) satisfies

T (z) = z E[T (z) m ].
For the geometric distribution, when p > 0.5 we have

T (z) = 1 -1 -4(1 -p)pz 2p (75)
and T (1) = 1-p p is the probability to have a finite tree. For the shifted geometric distribution we simply have T (z) = 0. If we omit the variation of the fraction of discovered vertices θ, the tree search experience starts by a sequence of finite trees, before entering into the infinite tree. Said in other words, the depth reaches the level 1 and will never come back to zero. But it will hit level 1 a finite time before leaving it for ever. If the degree of the root is m, the search is limited to m subtrees among them there are one or more infinite trees. The probability that the first infinite sub-tree is the kth sub-tree is T (1) k-1 (1 -T (1)), therefore the cumulated size of the finite trees before the first infinite subtree is 1-T (z) m 1-T (z) (1 -T (1)), multiplying by z we get the p.g.f. of the duration to reach the ultimate level 2 (i.e. the level which will never be decremented afterward). Averaging over the distribution of degrees and normalizing on the fact that there is always an infinite sub-tree we get

z-z E[T (z) m ] 1-T (z) . Since z E[T (z) m ] = T (z) we get z-T (z)
1-T (z) whose average obtained by the first derivative at z = 1 is exactly 1 1-T (1) . The average duration from the ultimate level 2 to the ultimate is also

1-T (1) . Therefore we have E[d(t)] = (1 -T (1))t(1 + O(θ)) (76)
the term O(θ) comes from the impact of the variation of θ when t is small which have neglected in the first place. □ However we don't know how to go further. If we want to get back to a Markov chain we must replace the depth by the stack index I(t). The DFS can be regarded as keeping a stack of unexplored arcs, for which we have seen the start vertex but not the end. The stack evolves as follows: stack1

(1) If the stack is empty, pick a new vertex v that has not been seen before (if there is no such vertex, we have finished). Otherwise, pop the last arc from the stack, and reveal its endpoint v (which is uniformly random over all vertices). If v already is seen, repeat. (2) (v is now a new vertex) Reveal the outdegree m of v and add to the stack m new arcs, with unspecified endpoints. GOTO 1.

Let again v t be the tth vertex seen by the DFS, and let I(t) be the size of the stack when v(t) is found (but before we add the arcs from v t ). Also let η t be the outdegree of v t . Then I(1) = 0 and, in analogy with (5),

I(t + 1) = I(t) + η t -1 -ξ t + , 1 ≤ t < n, (77) It
where ξ t is the number of arcs leading to already seen vertices before we find a new one; we have P(ξ = k) = (1 -t n )( t n ) k and thus ξ t ∼ Ge(1 -t/n). In analogy with (7), we define also

I(t) := t-1 i=1 (η i -1 -ξ i ) = t-1 i=1 ζ i , (78) tIt
where we define ζ t := η t -ξ t -1. Then, as in (8),

I(t) = I(t) -min 1≤j≤t I(j). ( 79
) ItI
Note that

E ζ t = E η t -E ξ t -1 = λ - t/n 1 -t/n -1 = λ - 1 1 -t/n . (80)
Hence, uniformly in t/n ≤ θ * for any fixed θ * < 1,

E I(t) = t-1 i=1 E ζ i = (t -1)λ - t-1 i=1 1 1 -t/n = n ι(t/n) + O(1), (81)
where again θ 1 is the largest root in [0, 1) of ( 16), now with λ = E η 1 , the mean of P. The proof of Theorem 2.2 applies with very minor differences, and yields: TG1 Theorem 4.3. Suppose that the outdegree distribution has finite variance. Then

max 1≤t≤n I(t) -n ι + (t/n) = O L 2 (n 1/2 ). (84)
Moreover, v t+1 is a root if and only if I(t) + ζ t = I(t) + η 1 -1 -ξ t < 0, cf. (77). The arguments in the proof of Theorem 2.5 apply with minor differences, and show: Theorem 4.4. Theorems 2.5 and 2.6 hold for any outdegree distribution with finite variance, with ρ := 1 -θ 1 -λ 2 (1 -θ 1 ) 2 .

The previous results suggest that the E I(t) and E d(t) asymptotically are just proportional by a fixed factor independent of t; note that both have the same root θ 1 . We make the following conjecture. ]. Remark 4.6. The quantity z 1 is the probability that a Galton-Watson tree generated by the outdegree distribution η is finite. The reason to invoke this probability is that E d(t) should be asymptotically equivalent to θ times the probability of an infinite tree (1 -z 1 ) when θ → 0. For the geometric distribution the probability is 1-p p thus with λ = p 1-p the proportional factor is 1-p p and we find again our first result. When λ < 1 the fixed point z 1 > 1 and the proportional factor remains positive. For the shifted geometric distribution the Galton-Watson tree is always in finite, thus z 1 = 0 and the proportional factor is again 1-p p .

Figure 1 .

 1 Figure 1. Example of a depth-first forest (jungle) from [1], by courtesy of Donald Knuth. Tree arcs are solid (e.g. 9○→ 3 ○). For example, 3○ 3 ○ is a loop, 2 ○ 3 ○ is a back arc, 9 ○ 7 ○ is a forward arc, 8 ○ 4○ and 0 ○ 2 ○ are cross arcs.

1 .

 1 Figure 1. Example of a depth-first forest (jungle) from [1], by courtesy of Donald Knuth. Tree arcs are solid (e.g. 9○→ 3 ○). For example, 3○ 3 ○ is a loop, 2 ○ 3 ○ is a back arc, 9 ○ 7 ○ is a forward arc, 8 ○ 4○ and 0 ○ 2 ○ are cross arcs.

  ) m* For t ≤ T * , the definition of M * in (25) implies min 1≤j≤t d(j) -n min 1≤j≤t ℓ(j/n) ≤ M * .(26) gal

  (59) follows by Corollary 2.4. C: We have L + T + B + F + C = M , and thus (60) follows from (56) and (57)-(59), noting that (61)-(62) imply β + χ = λ/2, and thus λ -(χ + 2β) = χ. □

1 1 -

 1 T[START_REF] Knuth | The Art of Computer Programming[END_REF] . And so forth to reach the ultimate level k we need an average duration

  k

  dτ = λθ + log(1 -θ). (82) Let ι + (θ) := [ ι(θ)] + = λθ + log(1 -θ), 0 ≤ θ ≤ θ 1 , 0, θ 1 ≤ θ ≤ 1,

Conjecture 4. 5 .

 5 If λ > 1, then, for all t ∈ [0, 1], we have E d(t) = 1-z1 λ-1 E I(t) + o(n) where z 1 ∈ [0, 1) satisfies z 1 = E[z η 1

Supported by the Knut and Alice Wallenberg Foundation.

Inria Saclay Ile de France, France Email address: philippe.jacquet@inria.fr Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden Email address: svante.janson@math.uu.se URL: http://www2.math.uu.se/∼svante/


Chang Liu

Wen-Shen Liu

Hermine Huot

Mei-Na Guo

Shi-Chen Zhu

Hong-Xiang Zheng

Jean Louis Morel

Ye-Tao Tang

Rong-Liang Qiu

Boehmeria Nivea

Biogeochemical cycles of nutrients, rare earth elements (REEs) and Al in soil-plant system in ion-adsorption REE mine tailings remediated with amendment

Keywords: fiber plant, mobilization, rhizosphere, root characteristic

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Rare earth elements (REEs) are a group of metallic elements comprised of fifteen lanthanide elements plus yttrium. The lanthanides are typically split into two subgroups, light REEs (LREEs, La to Eu) and heavy REEs (HREEs, Gd to Lu) [START_REF] Tyler | Rare earth elements in soil and plant systems -A review[END_REF]).

As the demand of REEs increases globally, the mining of REE deposits has rapidly grown [START_REF] Yang | China's ionadsorption rare earth resources, mining consequences and preservation[END_REF]. Ion-adsorption REE deposits, resulting from the weathering of granite under (sub)tropical climate, are mainly located in South China, and later found in Vietnam, Myanmar and Madagascar [START_REF] Borst | Adsorption of rare earth elements in regolith-hosted clay deposits[END_REF]. In these deposits, most REEs exist as trivalent ions and are adsorbed onto the clay minerals (e.g. kaolinite) and Fe/Mn oxides originated from the granite. Thus, the REEs can be easily leached out through a simple ion-exchange process using a salt solution: 2(Kaolin) 3-RE 3+ + 3(NH4)2SO4 → 2(Kaolin) 3-(NH4) + 3 + RE 3+ 2 (SO4) 2- 3 . The heap leaching technology used for decades in South China has consisted of: i) removing the vegetation and the topsoil;

ii) excavating the REE-rich clays (ores) from the (semi)weathered layers at a depth of 5-20 m; iii) pilling up the ores into a heap on a plastic sheet and leaching them with ammonium sulfate solution; iv) recovering REEs from the leachates [START_REF] Xiao | Progress in the green extraction technology for rare earth from ion-adsorption type rare earths ore[END_REF].

This mining technology has induced many environmental damages, such as water and soil erosion, vegetation destruction and mine tailings generation [START_REF] Yang | China's ionadsorption rare earth resources, mining consequences and preservation[END_REF]. For instance, it was reported that the area of REE mine tailings in Ganzhou city in the Jiangxi province exceeded 100 km 2 [START_REF] Liu | Limiting factors for restoration of dumping sites of ionic rare earth mine tailings[END_REF].

Though some pioneer plants have begun to settle after several years of abandonment, the revegetation of these mine tailings under natural succession is quite slow [START_REF] Liu | Limiting factors for restoration of dumping sites of ionic rare earth mine tailings[END_REF]. The poor physical conditions with soil compaction, low pH (4-5), low contents of nutrients (especially P) and organic matter (~0.1%), and low microbial activities are considered as the main obstacles for plant colonization on these mine tailings [START_REF] Liu | Limiting factors for restoration of dumping sites of ionic rare earth mine tailings[END_REF]. Moreover, these tailings still contain high residual REE concentrations [START_REF] Chao | Structure, variation, and co-occurrence of soil microbial communities in abandoned sites of a rare earth elements mine[END_REF], which are highly exchangeable even after 10 years of abandonment (>20% of the total REEs extracted with 1.0 mol/L MgCl2) [START_REF] Wen | Effect of simulated acid rain on soil acidification and rare earth elements leaching loss in soils of rare earth mining area in southern Jiangxi Province of China[END_REF]. This part of REEs may leach out and enter into the surrounding rivers and farmland soils, posing potential risks to local living organisms [START_REF] Li | A human health risk assessment of rare earth elements in soil and vegetables from a mining area in Fujian Province, Southeast China[END_REF]. Besides, the exchangeable Al 3+ concentration in the tailings soil is high and accounts for 80% of total cation exchange capacity (CEC) (Liu et al. 2020a), which may also inhibit plant colonization. Up to now, the phytoremediation field trials at these mine tailings have mainly focused on the restoration of soil fertility and microbial communities [START_REF] Zhou | Restoration of rare earth mine areas: organic amendments and phytoremediation[END_REF][START_REF] Liu | The genetic diversity of soil bacteria affected by phytoremediation in a typical barren rare earth mined site of South China[END_REF][START_REF] Wei | The effects of phytoremediation on soil bacterial communities in an abandoned mine site of rare earth elements[END_REF]Liu et al. 2020b), whilst the control of REE and Al availabilities and the cycling of REE and Al are largely ignored.

Phytostabilization, one strategy of phytoremediation, aims to establish a plant cover on the surface of contaminated soils, hence retaining contaminants within the vadose zone through root accumulation or precipitation of contaminants within the rhizosphere. This strategy can also reduce contamination dispersion by wind, water, and direct contact with humans or animals [START_REF] Bolan | Phytostabilization: a green approach to contaminant containment[END_REF]. Addition of organic amendments is a common practice in phytostabilization, which can improve soil structure, alleviate soil nutrient deficiency, and effectively stabilize contaminants in mine tailings [START_REF] Meeinkuirt | Influence of organic amendments on phytostabilization of Cd-contaminated soil by Eucalyptus camaldulensis[END_REF]. A recent pot experiment showed that organic amendments are effective in increasing the soil pH and organic matter concentration, thus reducing the REE concentrations in tailing soil solution (Liu et al. 2020a). However, organic amendment would induce an enrichment of dissolved organic carbon (DOC) in soil solution. As a result, heavy metals that are adsorbed on soil particles can be complexed by DOC and solubilized into the soil solution. For example, it was found that manure amendment was less efficient to stabilize heavy metals (e.g. Cd and Zn) [START_REF] Houben | Heavy metal immobilization by cost-effective amendments in a contaminated soil: Effects on metal leaching and phytoavailability[END_REF]).

Nevertheless, the effectiveness of phytostabilization practice with organic amendment in stabilizing REE and Al on the mine tailings needs further examination in the field.

Root development and activities are critical for the growth of plant and the cycling of nutrients and pollutants in the soil-plant system since roots are the interface between the plant and the soil environment [START_REF] Abe | Roots: The dynamic interface between plants and the earth[END_REF]. Plants can adapt to harsh soil condition through the manipulation of root morphology and metabolites [START_REF] Sasse | Multilab EcoFAB study shows highly reproducible physiology and depletion of soil metabolites by a model grass[END_REF]. Root morphological architectures and physiological adaptations are easily altered by nutritional stress [START_REF] Lambers | Root Physiology: from Gene to Function[END_REF]. A number of studies found that the changes of root morphology (e.g. fine root length) are critical for plant nutrient acquisition [START_REF] Hill | Morphology and response of roots of pasture species to phosphorus and nitrogen nutrition[END_REF][START_REF] Christopher | Root secondary growth: an unexplored component of soil resource acquisition[END_REF]. Plant root exudates, consisting of a complex mixture of metabolites, play a fundamental role in regulation of soil nutrient availability and detoxification of undesirable metal pollutants for plants [START_REF] Dakora | Root exudates as mediators of mineral acquisition in low-nutrient environments[END_REF][START_REF] Chen | Role of root exudates in metal acquisition and tolerance[END_REF]. Under conditions of nutrient deficiency (e.g. P), plant roots would release carboxylates and lower soil pH, thus solubilizing P, and Mn (or heavy metals) accidently [START_REF] Lambers | Leaf manganese accumulation and phosphorus-acquisition efficiency[END_REF]. Although the availability of heavy metals can be largely reduced in the early stage of phytostabilization practices, a subsequent re-mobilization of heavy metals may happen as a result of root secretion [START_REF] Pierzynski | Phytostabilization of metal mine tailings using tall fescue[END_REF][START_REF] Cui | Study on the remediation of heavy metal Cu and Cd contaminated soil by different ameliorators[END_REF]. However, there are no detailed studies reporting the in situ effects of root activities on soil properties and on the change of REE and Al availabilities in mine tailings.

The cultivation of economic crops in the mine tailings can produce biomass of industrial interest (e.g. fibers and bioenergy) and contribute to the socio-economic redevelopment of these mining areas [START_REF] Ding | Ecosystem services provided by heavy metal-contaminated soils in China[END_REF]. However, most studies focused on native or pioneer plants [START_REF] Wang | A review on in situ phytoremediation of mine tailings[END_REF], and the potential use of the economic crops for the remediation of mine sites is largely ignored. The fiber plant ramie (Boehmeria nivea L.) is a perennial herb, with a complex underground system composed of fine fibrous roots, radish roots and rhizome. In previous studies, ramie is regarded as a multi-metal tolerant plant, with strong tolerance to Pb, Zn, Cd, As, and has strong adaptability to harsh environment such as drought [START_REF] Yang | Constitutional tolerance to heavy metals of a fiber crop, ramie (Boehmeria nivea), and its potential usage[END_REF]). Ramie has also been found to have great REE tolerance and retain most of the REEs in roots under hydroponic culture (Liu et al. 2021a). Therefore, it could be a potential candidate for the phytoremediation of ion-adsorption REE mine tailings and for the production of fibers of economic value.

In this work, a field trial of phytostabilization combining addition of soil amendment and cultivation of the fiber plant ramie were conducted on ion-adsorption REE mine tailings in South China with the aim of investigating: (i) the effects of phytostabilization practice on the changes of soil physico-chemical properties, including nutrients, REE and Al availabilities; (ii) the root development and the role of root activities in the changes of nutrients, REE and Al availabilities; (ⅲ) the distribution, transfer and fractionation of REEs in the soil-plant system.

Materials and Methods

Site description

The study area is located in the Dingnan county of Ganzhou city, Jiangxi province, China (latitude: 24.99°, longitude: 115.05°). The terrain is hilly with an altitude ranging between 200 and 1200 m. It is situated in the subtropical monsoon humid climate zone with an average annual temperature of 18.9 ℃ and annual rainfall of 1620 mm [START_REF] Shi | Water pollution status and nitrogen pollution distribution patterns around ion absorbed rare earth tailings in Dingnan county, Jiangxi province[END_REF]. Dingnan is famous for its abundant reserve of ion-adsorption REE deposits, which have been mined since the 1980s. The studied REE mine tailings were resulted from the exploitation of REE deposits by heap leaching technology until 2008. The tailings were further terraced to protect them from landslides in 2012 (Fig. 1a).

The studied mine tailings covered a surface area of about 0.53 km², some parts of which were featured with steep slopes (> 45°) and subjected to strong erosion (Fig. 1). In 2015, the vegetation coverage was limited (< 5%) and composed mainly of pioneer species (e.g. Miscanthus sinensis, Pinus massoniana). Since 2015, this site has been used as a demonstration base to implement remediation projects using amendments and different plants.

Field plot description and samplings

Remediation practices

Two field plots (zones A and B) were set at the demonstration base in July 2015 to test the feasibility of cultivating fiber plants on ion-adsorption mine tailings (Fig. 1).

Different remediation practices in terms of fertilization, tillage and planting have been applied in both plots (about 2000 m 2 for each). Before planting, organic amendment (pig manure and sawdust with a ratio of 2:1, v/v) was added at a rate of 5-6 kg/m 2 and mixed with the first 10-12 cm of the tailings by tillage. Based on laboratory experiments, this organic amendment was shown to be efficient in supporting the growth of fiber plants in these mine tailings. Compared to manure, the mixture of manure and sawdust showed a slower release of DOC and nutrients [START_REF] Chen | Effects of organic amendments and biochar on ecological remediation of ionic rare earth mine tailings[END_REF].

Plant resources (seeds and seedlings) for cultivation were brought from the Institute of Bast Fiber Crops (Changsha, Chinese Academy of Agricultural Sciences).

In the zone A, an annual fiber plant was sown with a density of 3 g/m² in July 2015. In June 2016, plants were removed, and organic amendment was further added at a rate of 5-6 kg/m² and mixed in the first 10-12 cm. And then, one-year ramie seedlings of about 35 cm in height (cultivar Zhongzhu No.1), were planted in rows (3-5 plants per m²). In the zone B, ramie seedlings (one-year plants, about 35 cm in height) were planted in rows (3-5 plants per m²) in July 2015. In June 2016, the aboveground parts of ramie plants were cut and granules of inorganic fertilizers (N/P/K) were introduced at a rate of 30-45 g/m 2 .

Soil and plant sampling

The sampling of both field plots was performed in November 2017. For each zone (A and B), five whole ramie plants with soils associated to roots were sampled.

Rhizosphere soils were separated into two fractions: (i) the soil loosely adhering to the roots (LAR) which was collected by shaking the roots for 3 min until almost no soil particles dropped and (ii) the remaining soil tightly adhering to the roots (TAR) which was then recovered by brushing the roots [START_REF] Otero | Micronutrients and toxic trace metals in the bulk and rhizosphere soil of the spontaneous vegetation at an abandoned copper mine in Galicia (NW Spain)[END_REF]. For comparison, bulk soils were collected between two plants in each zone. In addition, bare tailing samples were collected at three locations situated near the field plots and were considered as controls without amendment and planting (Fig. 1b).

Analyses Preparation and analysis of soil samples

Soil samples were air-dried for two weeks and sieved through a mesh of 2 mm. Roots and stones were discarded from soils. An aliquot of 2-mm sieved soil sample was further ground (<0.15 mm) for the elemental analysis. The total carbon (TC) and nitrogen (TN) were analyzed by dry combustion using an elemental analyzer (Elemental, Vario EL, Germany). The concentrations of soil total REEs and other major and trace elements (e.g. Al, Fe, Ca, P, Mn) were measured at ALS Chemex (Guangzhou)

Co., Ltd. Namely, about 1 g of the air-dried soil was digested by lithium borate fusion method and analyzed by inductively coupled plasma mass spectrometry (ICP-MS) (Agilent VISTA, USA) for REEs and X-ray fluorescence spectrometry (XRF) (PANalytical PW2424, Netherlands) for major elements (Liu et al. 2020a).

The soil pH was measured at a ratio of 1: 2.5 (soil: deionized water, w/v) (Mettler Toledo, LE438 pH, Germany). An extraction using 0.001 mol/L CaCl2 solution with a ratio of 1: 2 (soil: CaCl2 solution, w/v) was performed to mimic the soil solution [START_REF] Groenenberg | Transfer functions for solidsolution partitioning of cadmium, copper, nickel, lead and zinc in soils: derivation of relationships for free metal ion activities and validation with independent data[END_REF]. Namely, 20 g of air-dried and < 2 mm soil was mixed with 40 mL 0.001 mol/L CaCl2 solution. The suspensions were shaken horizontally at 80 rpm for 48 h. After centrifugation and filtration through a 0.45-μm membrane, the extracts were analyzed for the electrical conductivity (EC) using an electrochemical analyzer (INESA DDS-307A, China), for the CaCl2-extractable concentrations of organic carbon (CaCl2-EOC) using a TOC analyzer (Elemental, Vario TOC, Germany) and for the CaCl2-extractable concentrations of major and trace elements at ALS Chemex (Guangzhou) Co., Ltd using ICP-MS and inductively coupled plasma optical emission spectrometry (ICP-OES) (Agilent VISTA, USA).

Preparation and analysis of plant samples

For plant sample analyses, each plant was separated into leaves, stems, fine roots (0-5 mm in diameter) and thick roots (>5 mm, including radish roots and rhizomes) (Fig. 1d) [START_REF] Boehm | Methods of Studying Root Systems[END_REF][START_REF] Bai | Research progress of ramie root systems[END_REF]. In order to eliminate the dust from the surface of the plants, and the soil particles adhering to the roots, each plant part was carefully washed with tap water and deionized water. After cleaning, the fine roots were scanned (Epson perfection, V700 photo, Indonesia) and analyzed using a WinRHIZO software to measure the fine root total length (TL), surface area (SA), average diameter (AD), root volume (RV), root length depending on the diameter classes (0-0.2 mm, 0.2-1 mm, 1-2 mm, 2-3 mm, 3-4 mm, 4-5 mm). The small fine root (<1 mm) and fine root (<5 mm) morphological parameters are defined as follows:

𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑟𝑜𝑜𝑡 𝑙𝑒𝑛𝑔𝑡ℎ (𝑆𝑅𝐿, 𝑐𝑚/𝑔) = fine root length (cm) fine root biomass (g) 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑟𝑜𝑜𝑡 𝑠𝑢𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 (𝑆𝑅𝐴, 𝑐𝑚/𝑔) = fine root surface area (cm 2 ) fine root biomass (g) 𝑠𝑚𝑎𝑙𝑙 𝑓𝑖𝑛𝑒 𝑟𝑜𝑜𝑡 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 (𝑆𝐹𝑅𝑃, %) = small fine root length (cm) fine root length (cm) × 100

𝑟𝑜𝑜𝑡 𝑡𝑖𝑠𝑠𝑢𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 (𝑅𝑇𝐷, 𝑔/cm 3 ) = fine root biomass (g) fine root volume (cm 3 ) Plant materials were dried at 105 ℃ for 30 min and 60 ℃ for 48 h and then ground to <1 mm and preserved in polyethylene bags. The plant samples were digested by HNO3 (65%, 6 ml) and HCl (37%, 2 ml). Afterwards, the trace and major elements in the digested solutions were analyzed by ICP-MS and ICP-OES at ALS Chemex (Guangzhou) Co., Ltd as described above.

REE fractionation analysis

The REE fractionation in the soil-plant system results from the small variations in the behaviors of individual REEs during soil and plant physiological processes [START_REF] Tyler | Rare earth elements in soil and plant systems -A review[END_REF]). The Ce anomaly and REE fractionation pattern (i.e. LREE or HREE enrichment) are widely used indicators in the soil-plant system. The transfer factors (Kn), which are defined as the REE concentrations in one compartment divided by the concentrations in another compartment, were also calculated along the REE transfer route. The REE transfers in the soil-ramie system can be divided into 5 steps: I, from rhizosphere soil (TAR) to soil solution; II, from rhizosphere soil solution to fine roots; III, from fine roots to thick roots; IV, from thick roots to stems; V, from stems to leaves. The transfer factors of these steps are calculated as follows: The REE fractionation during each transfer process can be roughly estimated by (La/Yb), and Ce anomaly.

𝐾 𝐼 = [REE]
𝐾 𝑛 ( * 𝐶𝑒) = 2 × K 𝑛 (Ce) K 𝑛 (La) + K 𝑛 (Pr) 𝐾 𝑛 (𝐿𝑎/𝑌𝑏) = K 𝑛 (La) K 𝑛 (Yb)
The Ce anomaly value (*Ce) >1 and <1 means positive and negative anomaly, respectively, and the ratio of La to Yb >1 and <1 means a LREE and HREE enrichment pattern, respectively [START_REF] Wei | Rare earth elements in naturally grown fern Dicranopteris linearis in relation to their variation in soils in South-Jiangxi region (Southern China)[END_REF]Liu et al. 2021b).

Data analysis

The SPSS 20.0 statistical analysis software was used for data analysis. All results were presented as arithmetic means with standard deviation calculated on three or five samples. Soil physico-chemical properties, plant growth characteristics, element concentrations in plant organs were compared using one-way analysis of variance (ANOVA) followed by a Duncan test at a significance level of 0.05. For one parameter, values with different letters refer to significant differences between the different samples from zones A and B and the controls. For some parameters, if the data were not normally distributed, they were transformed by normal score to follow a normal distribution before being tested. Matrices of correlation were performed on the soil physico-chemical properties and plant elemental composition or root morphological characteristics using a Pearson correlation two-sided test.

Results

Bulk and rhizosphere soil properties

The general physico-chemical characteristics of the bare tailings, bulk and rhizosphere (LAR and TAR) soils are shown in Fig. 2. The pH values of the mine tailings after planting and amendments (4.80-6.00) were significantly higher than those of the bare tailings (4.38-4.47). There was no significant difference in pH between the bulk, LAR and TAR soils in both zones A and B. The TC and TN concentrations showed significant differences between different soil fractions, and were in the order of bare tailings < bulk soil < LAR < TAR in both the zones A and B. Whatever the soil fraction, the concentrations of TC and TN were higher in the zone A compared with the zone B. The total P concentrations were higher in the LAR (30-100 mg/kg) and TAR (20-120 mg/kg) soils than in the corresponding bulk soil (20-80 mg/kg), and much higher than that in bare tailings (4.36 mg/kg). The total concentrations of other nutrients (Ca, K) showed similar trends as the total P concentrations (Table S1). Concerning the bulk soils, some chemical properties (pH, EC and total N, P, Ca, K, REE concentrations) were significantly higher in the zone A than in the zone B.

The reclaimed tailings generally presented higher EC values in soil CaCl2 extracts (154-693 μS/cm) than the bare tailings (185-200 μS/cm). For each zone, the EC values of rhizosphere soils were higher than the corresponding bulk soils. The concentrations of soil CaCl2-EOC showed significant differences between the soil fractions and were in the order of bare tailings < bulk soil < LAR < TAR in both the zones A and B. The concentrations of soil CaCl2-EOC were higher in the zone A compared with the those in the zone B for the corresponding soil fractions. The extractable nutrient concentrations (K, Mg, Mn, Na, S, Zn) were significantly higher in the rhizosphere soils (LAR and TAR fractions) compared with the corresponding bulk soils and the bare tailings (Table S2).

The total REE concentrations varied in the bare tailings (281-729 mg/kg) and the reclaimed tailings (zone A: 926-3460 mg/kg; zone B: 551-1030 mg/kg), revealing the heterogeneity of the distribution of residual REEs in the tailings. The total Al concentrations were around 8.5% in all the soil samples. The CaCl2-extractable REE concentrations in the bare tailings (controls) were ten times higher than those in the reclaimed tailings. The proportions of extractable REE concentrations out of the total REE concentrations in the soils followed the order of control (0.11-0.79%) > zone B bulk soil (0.04-0.13%) > zone A bulk soil (0.01-0.06%). Similar to those of REEs, the extractable Al concentrations and the proportions of extractable Al out of the total Al concentrations were both ten times higher in the controls than in the reclaimed tailings.

No significative difference in the extractable REE and Al concentrations or proportions was found between bulk and rhizosphere soils in both zones A and B (Fig. 3).

Plant growth and elemental distribution

The biomass and root characteristics of ramie plants are shown in Fig. 4 and Table S3.

Among the different ramie organs, the thick roots showed the highest dry biomass both in the zone A (159-406 g per plant) and the zone B (251-1090 g per plant). The thick roots accounted for 85-89% of the total dry biomass of ramie, followed by the stems (9.6-14.4%), fine roots (1.3-3.0%) and leaves (0.02-0.12%). Each ramie plant had an average ramet number of 6 in both zones. The average height of these ramet stems was 170 and 163 cm in the zones A and B, respectively.

The characteristics of the fine roots (< 5 mm in diameter) of ramie plants are presented in Table A3. The root morphological parameter AD was higher in the zone B than the zone A, while the TL, SA, and RV of the fine roots showed no significant difference between the zones A and B. The length of the small fine roots (<1 mm in diameter)

represented more than 99% of the total fine root length, including 27 % for the very fine roots with a diameter <0.2 mm. The SRL was significantly higher in the zone A than that in the zone B, while the SRA and RTD were similar in both zones.

The distribution of nutrients varied greatly between the ramie organs and between the zones A and B (Tables S4 andS5). accumulated by ramie plant was stored in the thick roots in the zones A and B (Fig. 5).

Transfer and fractionation of REEs in the soil-ramie system

The transfer factors in the soil-ramie system are shown in Fig. 6 and Table S6. The

Relationships between soil properties and plant characteristics

A correlation matrix analysis was performed on selected physico-chemical properties and extractable REEs of the bare tailings, the bulk and rhizosphere soils (Table 1).

Results showed that extractable REE concentrations were positively correlated with extractable Al concentrations, and negatively correlated with soil pH and soil total C, N, P and Ca concentrations (p<0.05).

A correlation matrix analysis was also performed on selected physico-chemical properties of TAR soils and plant parameters (Table 2 andTable S7). In plants, four root characteristics (length for the different diameter classes, TL, SA, RV) were positively correlated with REE, Al and nutrient (Ca, P) concentrations of fine roots (p<0.05). The amount of REEs accumulated in the whole plant was also positively correlated with these four root characteristics. Moreover, these four root characteristics were positively correlated with soil pH, TC, TP, and soil extractable Mg concentration. The SRL was positively correlated with pH, TC, and TP in soils, and the SRA was positively correlated with TC in soils. However, the AD and SFRP were positively and negatively correlated with the extractable REE concentrations in the soil, respectively.

Discussion

Effects of phytostabilization practices on dynamics of REEs, Al and nutrients in soil

In line with previous studies, our results confirmed that the soils of bare tailings exhibit low pH, low concentrations of CaCl2-EOC, TC, TN, and nutrients (Ca, Mg, P, S, etc.), but high concentrations of extractable REE and Al (Fig. 3). The REE concentrations in the tailings were higher than the background values in the soils of mainland China (188 mg/kg) or Jiangxi province (221 mg/kg) [START_REF] Wei | 15 Rare Earth Elements in Chinese Soil[END_REF], and showed high proportions of available REEs. A pot experiment has shown that plants can hardly grow on these tailings without soil improvement (Liu et al. 2020a). Our results showed that the application of organic amendment and planting with ramie resulted in an increase in soil pH, soil CaCl2-EOC, TC, TN and nutrient concentrations. The organic amendment is a source of organic matter as well as macronutrients and micronutrients, which are critical for supporting plants growth and adaptation in contaminated soils [START_REF] Meeinkuirt | Influence of organic amendments on phytostabilization of Cd-contaminated soil by Eucalyptus camaldulensis[END_REF]. The introduction of plant and organic amendment may also promote the formation of stable aggregates, providing space for root growth and for the survival and breeding of soil invertebrates and microorganisms [START_REF] Chiang | Low-molecular-weight organic acids and metal speciation in rhizosphere and bulk soils of a temperate rain forest in Chitou, Taiwan[END_REF]).

The soil pH was among the most obvious factors controlling the concentrations of extractable REE and Al in the mine tailing soils (Table 1). Under strong acid soil conditions in the bare tailings (pH 4.4), REEs and Al that adsorbed on clays, organic matter and Fe-/Mn-oxides may be released into the soil solution. With the increase of soil pH in the reclaimed soils, the REEs and Al in the soil solution would be precipitated by REE/Al hydroxides and Fe-/Mn-oxides (Dinali et al. 2004). Significant correlations between soil REE/Al and TC concentrations were also observed, indicating that REEs and Al could be bound by the functional groups (e.g. hydroxyl and carboxyl) of organic matter (Migaszewski and Gałuszka 2015). Moreover, with the increase of P concentrations, the REE and Al ions may also be precipitated by phosphates. The reasons why REE and Al showed a similar behavior could be that both REEs (except Ce and Eu in some conditions) and Al are trivalent elements, which show many similarities of chemical properties during the geochemical processes [START_REF] Pletnev | Classification of metal ions according to their complexing properties: a data-driven approach[END_REF]. Therefore, the phytostabilization practices with ramie planting and organic amendment can protect the plants from REEs and Al toxicity, and may reduce the risk of groundwater contamination and other biological exposure. The significant differences of soil pH, EC, TN, and soil total P, Ca, K, REE concentrations between the zones A and B may be ascribed to the different planting strategies and amendment additions. The higher percentages of extractable REE and Al in the soils of zone B than zone A may be due to the lower soil pH, and soil TC and P concentrations of zone B.

Root development and effects of root activities on dynamics of REEs, Al and nutrients in soil

Soil properties may constrain the plant and root development and the morphological characteristics of the root system. Matrices of correlation (Table 2) indicate that the beneficial soil physico-chemical properties (pH, TC and nutrients) promote the fine root growth of ramie, however, the potential toxic REE and Al have adverse effects on the fine root length. Previous studies have reported that the most obvious toxicity phenomenon of Al and REE to plant is the inhibition of root elongation and activities [START_REF] Tyler | Rare earth elements in soil and plant systems -A review[END_REF]Abdulaha-Al et al. 2018). A recent study found that the rice root morphological features (e.g. the number of lateral root of adventitious roots) were sensitive to high concentration of REE [START_REF] Shi | The accumulation and effect of rare earth element neodymium on the root of rice seedlings[END_REF]. The difference of age of ramie could also contribute to the differences in root morphology observed between the zones A and B. Physical properties could also be a limiting factor for root development in the mine tailings and should be taken into consideration in future studies.

In turn, the root activity and changes of the root morphological parameters may affect the soil properties in the rhizosphere. Significant enrichment of C and nutrients can be seen in the rhizosphere samples compared to bulk soil samples (Fig. 2), indicating that in addition to soil amendment, ramie planting could further improve the soil properties.

This enrichment of C and nutrients in rhizosphere soils has been previously reported for other mine soils phytoremediation practices by the planting of Digitaria sanguinalis, Picris hieracioides, Conyza albida, Setaria viridis [START_REF] Campillo-Cora | Copper and zinc in rhizosphere soil of wild plants growing in long-term acid vineyard soils. Insights on availability and metal remediation[END_REF]). On the one hand, plant roots can directly release exudates or protons into the rhizosphere zone, thus influencing nutrient mobility and bioavailability, and other physico-chemical properties [START_REF] Hinsinger | Rhizosphere geometry and heterogeneity arising from root-mediated physical and chemical processes[END_REF][START_REF] Qasim | Rhizosphere effects of Populus euramericana Dorskamp on the mobility of Zn, Pb and Cd in contaminated technosols[END_REF]. In fact, ramie has been found to enhance soil fertility by fine root turnover and root exudation [START_REF] Bene | Changes in soil chemical parameters and organic matter balance after 13 years of ramie[END_REF]). On the other hand, the plant would recruit certain beneficial microbes (e.g. P solubilization microbes) in the rhizosphere under nutrient deficiency, thus indirectly increasing nutrient availabilities [START_REF] Alori | Microbial phosphorus solubilization and its potential for use in sustainable agriculture[END_REF]. Our former study found that the combination of amendment and ramie planting treatment recruited much more beneficial microbes compared to the only amendment treatment (Liu et al. 2020b).

A number of studies reported that the planting induces strong rhizosphere effects, which finally induces an increase of available REEs in the soils [START_REF] Wang | Comparison of speciation and bioavailability of rare earth elements between wet rhizosphere soil and air-dried bulk soil[END_REF][START_REF] Wiche | Interspecific root interactions between white lupin and barley enhance the uptake of rare earth elements (REEs) and nutrients in shoots of barley[END_REF]. Compared to no planting treatment, it was found that planting of a REE hyperaccumulator Phytolacca americana significantly decreased soil pH and increased extracted REE concentrations of the REE mine tailings (Liu et al. 2020a). However, in the present study, no significant differences of extractable REE and Al concentrations were found between the rhizosphere and bulk soil samples (Fig. 5). One possible reason is that the rhizosphere effect did not induce a significant decrease of soil pH (Fig. 2a).

Another possible reason is that ramie absorbed high amount of REEs and Al from the soil solution and retained them within the roots. Moreover, the REEs show high affinities to oxalate (a root exudate for many plants), forming stable REE-complexes or even precipitates that are insoluble [START_REF] He | One-step separation and recovery of rare earth and iron from NdFeB slurry via phosphoric acid leaching[END_REF].

REE accumulation and fractionation of soil-ramie system

In this study, ramie presented large biomass in underground parts, especially the thick roots (Fig. 4). Consequently, the quantity of REEs and Al accumulated by the plant was mostly retained within the thick roots for both zones. The thick roots are a "storehouse", which can store a series of nutrients (e.g. N, P, K, etc.) (Table S5) [START_REF] Bai | Research progress of ramie root systems[END_REF].

Ramie might use these nutrients for next growth season, especially under nutrient deficiency [START_REF] Wu | Seasonal nutrient cycling and enrichment of nutrient-related soil microbes aid in the adaptation of ramie (Boehmeria nivea L.) to nutrient-deficient conditions[END_REF]. Most of the REEs and Al in the plant roots are usually tightly complexed by the cell walls (e.g. REE hyperaccumulator Dicranopteris linearis and normal plant cucumber) [START_REF] Zhang | Comparative toxicity of nanoparticulate/bulk Yb2O3 and YbCl3 to cucumber (Cucumis sativus)[END_REF]Liu et al. 2021c). Thus, the stored REEs and Al in the thick roots of ramie are unlike to be redistributed in the plant any more.

enrichment of CeO2 precipitation on the root surface. These particles may further activate REE endocytosis at the root cells membrane [START_REF] Wang | Rare earth elements activate endocytosis in plant cells[END_REF]. [START_REF] Brioschi | Transfer of rare earth elements (REE) from natural soil to plant systems: Implications for the environmental availability of anthropogenic REE[END_REF]

Conclusion

The wide exploitation of ion-adsorption REE deposits in South China have led to a large area of mine tailings with limited remediation. In the present study, a 2-year field trial of phytostabilization combining addition of organic amendment and cultivation of ramie was conducted on a typical ion-adsorption REE mine tailing. Our results showed that this phytostabilization strategy not only improves soil properties, but also decreases the extractable concentrations of potentially toxic elements (REEs and Al). By comparing the soil physico-chemical properties between the rhizosphere soils and the corresponding bulk soils, our study also showed that ramie planting and the root activities further improve soil properties through C input and nutrient mobilization.

Moreover, this fiber plant succeeded to grow on the amended mine tailings and the major quantity of REEs taken up by the plant were restricted within the thick roots. In sum, we conclude that phytostabilization practice with organic amendment and ramie cultivation is a promising strategy for the remediation of ion-adsorption REE mine tailings in South China. The values in the matrix are the Pearson correlation coefficient (R) between element concentrations. The higher the R value is, the higher degree of correlation (positive or negative) between the two physico-chemical properties. The * refers to the significance level of the correlation (p-value), * p<0.05.

E: CaCl2-extractable concentration The values in the matrix are the Pearson correlation coefficient (R) between element concentrations and plant characteristics. The higher the R value is, the higher degree of correlation (positive or negative) between the two physico-chemical properties. The * refers to the significance level of the correlation (p-value), * p<0.05.

E: CaCl2-extractable concentrations; L (0-0.2): root length for roots with a diameter 0.00<d<=0.20 mm; L (0-2): root length for roots with a diameter 0.00<d<=2.00 mm. TL, fine root length; SA, surface area; AD, average diameter; RV, volume (RV); SRL, specific root length; SRA, specific root surface area; SFRP, small fine root percentage.

  The major elements accumulated by ramie plants were Ca, K and P, and to a lesser extent Mg and Fe. Phosphorus (P) concentrations in the organs showed the order of leaves ≈ stems > thick roots > fine roots in the zone A, and leaves > thick roots > stems > fine roots in the zone B. The amount of P (elemental concentration × dry biomass) accumulated in the different organs of ramie generally showed the order of thick roots (76.6-91.7%) > stems (7.33-21.3%) > fine roots (0.54-2.60%) > leaves (0.02-0.14%) in both the zones A and B.The concentrations and amount of REEs and Al in plants are presented in Fig.5. The concentrations of REEs in the different organs of ramie were distributed in the order of fine roots (248-590 mg/kg) > leaves (160-312 mg/kg) > stems (36-172 mg/kg) > thick roots (31-59 mg/kg) in both the zones A and B. Despite the high REE concentrations in leaves, the amount of REEs accumulated in leaves was low, being 0.01-0.30 mg per plant for both the zones A and B, and only accounted for about 0.5% of the total amount of REEs accumulated in the whole plant. The amount stored in the thick roots reached 12.9-21.3 mg per plant, and represented approximately 67.0% and 80.2% of the total amount of REEs accumulated in the whole plant in the zones A and B, respectively. The concentrations of Al in different organs of ramie showed the order of leaves (2920-10800 mg/kg) > fine roots (1690-6300 mg/kg) > thick roots (495-3140 mg/kg) > stems (641-2050 mg/kg) in the zones A and B. The amount of Al accumulated in the different organs showed similar trends to those of REEs. Namely, most of the Al (67-94%)

  results showed that KII > KV > KIV > KIII >KI, indicating a REE accumulation in fine roots compared with the soil extractable REE fraction (step II) and a slight enrichment in the leaves compared with the stems (step V). The slopes of KI (La/Yb) and KV (La/Yb) were 0.4-0.5 in the zones A and B, indicating a selective transport of HREEs during soil-solution and stem-leaf transfers. While the slopes of KII (La/Yb) (~4) suggested a selective absorption of LREEs by ramie roots. The Ce anomaly values was <1 for the steps I and IV, and >1 for steps II, III and V. In addition, there were no significant differences of transfer factors and line slopes between the zones A and B.

  speculated that the REE-oxyhydroxide particles in plant xylem sap could lead to the positive Ce anomaly during the transport process.Though the HREE concentrations in the mine tailings are much lower than the LREEs, the HREEs are easier to migrate from soil into the soil solution, and transport from underground part to the shoots. Further studies should not ignore the potential environmental risks of HREEs. Based on the current field trial, high mechanical strength fibers with low REE concentrations have been extracted from the ramie plants grown on the tailings[START_REF] Segovia | Industrial ramie growing on reclaimed ion-adsorption rare earth elements mine tailings in southern China: defibration and fibers quality[END_REF]). However, further work is needed to ensure sustainable management of this crop and long-term REE/Al phytostabilization of the mine tailings.

Fig. 1 .

 1 Fig. 1. Location of the experiment plots in an ion-adsorption rare earth element mine

  Zone A: the tailings were amended and planted with an annual fiber plant in 2015; plants were then removed, and the tailings were amended again and planted with ramie in 2016; Zone B: the tailings were amended and planted with ramie in 2015; the aboveground parts of ramie plants were harvested and fertilizers (N/P/K) was added in 2016.

Fig. 2 .

 2 Fig. 2. Selected chemical properties in bulk and rhizosphere (LAR and TAR) soil

Fig. 3 .Fig. 5 .

 35 Fig. 3. CaCl2-extractable concentrations of rare earth elements (a) and of Al (b) in bulk

Fig. 6 .

 6 Fig. 6. Patterns of rare earth elements (REE) transfer factors in the soil-ramie system in

Table 1

 1 Correlation matrix of selected physico-chemical properties of bare and reclaimed tailings (bulk and rhizosphere soils)

		pH	EC	TC	TN	Fe	Al	Mn	K	Ca	P	Si	REEs	E-Al	E-REEs
	pH	1
	EC	0.75 *	1
	TC	0.77 *	0.85 *	1
	TN	0.75 *	0.87 *	0.99 *	1
	Fe	0.17	0.11	0.10	0.10	1
	Al	0.28	0.23	0.22	0.26	0.14	1
	Mn	0.59 *	0.67 *	0.58 *	0.59 *	0.27	0.46 *	1
	K	0.07	-0.14 0.14	0.09	-0.05 -0.00 -0.30 1
	Ca	0.90 *	0.91 *	0.89 *	0.90 *	0.14	0.29	0.66 *	-0.06 1
	P	0.83 *	0.90 *	0.90 *	0.90 *	0.14	0.27	0.70 *	-0.03 0.95 *	1
	Si	-0.70 * -0.72 * -0.79 * -0.80 * -0.24 -0.76 * -0.70 * -0.06 -0.78 *	-0.77 *	1
	REEs	0.76 *	0.78 *	0.64 *	0.67 *	0.09	0.50 *	0.74 *	-0.28 0.83 *	0.82 *	-0.74 *	1
	E-Al	-0.71 * -0.45 * -0.58 * -0.55 * -0.31 -0.31 -0.26 -0.28 -0.59 *	-0.59 *	0.62 *	-0.38 *	1
	E-REEs -0.69 * -0.39 * -0.55 * -0.52 * -0.20 -0.35 * -0.13 -0.50 * -0.54 *	-0.52 *	0.59 *	-0.34 *	0.93 *	1

Table 2

 2 Correlation matrices of TAR soil physico-chemical properties, fine root morphology indicators and composition, and plant REE

	accumulation
		Fine roots			Whole plant	TAR
		REE	P	Ca	Al	REE	pH	TC	P	E-Al	E-Mg	E-Si	E-REE
	TL	0.71 *	0.96 *	0.92 *	0.83 *	0.65 *	0.84 *	0.68 *	0.71 *	-0.37	0.82 *	0.65 *	-0.30
	SA	0.71 *	0.97 *	0.95 *	0.86 *	0.71 *	0.80 *	0.64 *	0.67 *	-0.32	0.80 *	0.60	-0.25
	RV	0.70 *	0.98 *	0.97 *	0.88 *	0.77 *	0.74 *	0.58	0.62	-0.25	0.76 *	0.53	-0.17
	AD	-0.31	-0.58	-0.46	-0.40	0.01	-0.79 *	-0.65 *	-0.66 *	0.71 *	-0.67 *	-0.80 *	0.72 *
	L (0-0.2) 0.69 *	0.89 *	0.82 *	0.80 *	0.55	0.81 *	0.62	0.67 *	-0.44	0.70 *	0.56	-0.49
	L (0-2)	0.70 *	0.96 *	0.93 *	0.81 *	0.67 *	0.82 *	0.68 *	0.71 *	-0.34	0.85 *	0.67 *	-0.22
	SRL	0.47	0.49	0.29	0.23	-0.10	0.73 *	0.77 *	0.73 *	-0.60	0.62	0.83 *	-0.58
	SRA	0.43	0.34	0.14	0.08	-0.16	0.54	0.67 *	0.62	-0.44	0.46	0.70 *	-0.43
	SFRP	-0.26	0.15	0.08	0.05	-0.29	0.26	0.11	0.20	-0.76 *	0.11	0.37	-0.84 *
	RTD	-0.01	-0.34	-0.08	0.19	0.18	-0.05	-0.31	-0.23	-0.05	-0.05	-0.24	-0.03

Acknowledgements

This work is supported by the International Young Scientists Fund of the National Natural Science Foundation of China [Grant No. 21850410450]; the Key R&D Program of China of Jiangxi province [Grant No. 20192ACB70016]; the National Natural Science Foundation of China (NSFC) [Grant No. 41771343]; the 111 project [Grant No. B18060]; the "2017-2019 Sino-French Cai Yuanpei Programme" [Grant No. 38896SC] and the PhD scholarship awarded to Liu Chang (No. 201700260091) by the China Scholarship Council. The authors are indebted to the LabEx Ressources21 for its financial support. The "International Joint Lab-ECOLAND" is acknowledged for supporting this work. The "Research on Environmental Pollution Control and Ecological Restoration (092018122466v1)" funded by Jiangxi Waydiam

Contrary to the "storehouse" role of the thick roots, the fine roots function as a "capturer" of nutrients and contaminants from the soil [START_REF] Bai | Research progress of ramie root systems[END_REF]. This can be explained by the positive correlations between the fine root characteristics and the root REE/Al/nutrient concentrations (Table 2).

With the increase of atomic number of REE ions, their ability to adsorb on clay minerals decreases gradually, while their ability to be complexed and leached out increases [START_REF] Tyler | Rare earth elements in soil and plant systems -A review[END_REF]. Therefore, soil-to-soil solution step showed a HREE enriched pattern (Fig. 6), which was similar to that of soil-D. linearis system [START_REF] Wei | Rare earth elements in naturally grown fern Dicranopteris linearis in relation to their variation in soils in South-Jiangxi region (Southern China)[END_REF]. The negative Ce anomaly during soil-to-soil solution process may be due to the fact that Ce mostly exists as Ce oxides in the soil [START_REF] Tyler | Rare earth elements in soil and plant systems -A review[END_REF]). The reason why soil solution-to-fine root process showed an LREE enriched pattern may be ascribed to the preferential uptake of LREE by plant root through Ca 2+ channel, while the uptake of HREE may be inhibited by the high concentration of Al [START_REF] Yuan | The accumulation and fractionation of rare earth elements in hydroponically grown Phytolacca americana L[END_REF]). The HREE enrichment in the leaves compared to the stems measured in ramie was also observed in the REE hyperaccumulator P. americana. A previous study carried out on this plant has revealed that organic ligands such as citric acid play an important role in long-distance transport of REEs in the xylem, leading to a preferential enrichment of HREE in the leaves [START_REF] Yuan | The accumulation and fractionation of rare earth elements in hydroponically grown Phytolacca americana L[END_REF]. The positive Ce anomalies during the uptake processes (solution-to-fine roots, fine root-to-thick roots, and stems-to-leaves) were also found in our former hydroponic experiment (Liu et al. 2021a). It may be ascribed to the preferential Environmental Protection Technology Co., Ltd is also acknowledged for supporting this work.

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: None.


Paul Clavin

Raúl Hernández Sánchez

Bruno Denet

Study of the direct initiation of spherical detonations

An analysis of the direct initiation of gaseous detonations in spherical geometry is presented, for a Mach number close to unity (small heat release). In this limit, the evolution is described by a Burgers equation, modified by a reaction and a curvature term. The solutions of this equation are presented in a pulsating detonation case.

INTRODUCTION

The direct initiation of a spherical detonation has attracted a lot of attention, starting with the suggestion of Zeldovich that a successful initiation requires that the time to reach the Chapman Jouguet (CJ) velocity is larger than the reaction time. This criterion leads to a critical radius of the order of the detonation thickness of the planar CJ detonation. It was shown experimentally that this critical radius is much too small. A quasi-steady analysis by He and Clavin [START_REF] He | On the direct initiation of gaseous detonations by an energy source[END_REF] (see also [START_REF] He | Theoretical determination of the critical conditions for the direct initiation of detonations in hydrogen-oxygen mixtures[END_REF]), using a square wave model, showed that the curvature of the detonation leads to a turning point representative of a curvature-induced quenching; the spherical CJ velocity has a C-shaped curve (see Figure 1) and the quasi-steady solutions exist above a critical radius much larger than the detonation thickness. However the quasi-steady approximation is not fully satisfactory and cannot reproduce the behavior of DNS trajectories, especially near criticality [START_REF] He | On the direct initiation of gaseous detonations by an energy source[END_REF], [START_REF] He | Theoretical determination of the critical conditions for the direct initiation of detonations in hydrogen-oxygen mixtures[END_REF], [START_REF] Eckett | The role of unsteadiness in direct initiation of gaseous detonations[END_REF]. A theoretical model with an inner structure more realistic than the square wave model, will be used to assess the role of the unsteady effects, in the case of successful initiations or failures.

MODEL AND RESULTS

In the limit of a small heat release, the full Euler equations reduce to a Burgers equation, modified by a reaction term and curvature term [START_REF] Clavin | Analytical study of the direct initiation of gaseous detonations for small heat release[END_REF][START_REF] Clavin | Asymptotic analysis of the critical dynamics of spherical gaseous detonations[END_REF]. This inner detonation structure has to be matched to a rarefaction wave. Sufficiently close to the Chapman-Jouguet solution, this Burgers equation can also be transformed in an integral equation [START_REF] Clavin | Analytical study of the direct initiation of gaseous detonations for small heat release[END_REF], of the type recently used in the plane case by the authors in [START_REF] Clavin | Decay of plane detonation waves to the self-propagating Chapman-Jouguet regime[END_REF].

We present here solutions of the reactive Burgers equation with curvature [START_REF] Clavin | Asymptotic analysis of the critical dynamics of spherical gaseous detonations[END_REF]. For a small heat release, this equation can be written:

∂µ ∂τ + µ - y b ∂µ ∂ξ = 1 2 ω(ξ, y(τ )) - (1 + µ) rf (τ ) , (1)
ξ = 0 : µ = 1 + 2y(τ )/b. ( 2
)
where µ is a reduced flow velocity in the laboratory frame, ω is the reaction term, the last term in the right hand side of (1) is a curvature term. Using the boundary condition at the shock (2), the reduced detonation velocity y can be obtained (y = 0 corresponds to the Chapman Jouguet velocity). An example of numerical integrations of this equation for different initial radii is shown in Figure 2.

CONCLUSIONS

For detonations close to the CJ velocity, it has been possible to simplify the full system of equations describing the spherical detonation to get a Burgers equation with reaction and curvature terms. Concerning the trajectories of the lead shock in direct initiation, the limit of small heat release is in qualitative agreement with direct numerical simulations, except for the overshoot during the re-acceleration phase which is sometimes observed in DNS.

2 Figure 1 :

 21 Figure 1: Sketch of the trajectories, x is proportional to the radius, y is proportional to the difference between the detonation velocity and the plane CJ velocity. Dotted lines represent schematically unsteady trajectories obtained by DNS. Solid lines are the quasi-steady trajectories.

Figure 2 :

 2 Figure 2: Examples of numerical integration of the reactive Burgers equation for a weakly unstable detonation b = 2. The failure occurs for initial radius small enough rfi < 1.5 because of chemical-kinetics quenching.

* Corresponding author. E-mail: paul.clavin@univ-amu.fr

Acknowledgements: We thank Prof. Amable Liñan for stimulating discussions. This work was supported by the French National Research Agency under agreement ANR-18-CE05-0030,