We then apply this result to give the limit in distribution of a critical Galton–Watson tree conditioned on having a large number of protected nodes

©Applied Probability Trust2017

LOCAL LIMITS OF GALTON–WATSON TREES CONDITIONED ON THE NUMBER OF

PROTECTED NODES

ROMAIN ABRAHAM,^∗Université d’Orléans AYMEN BOUAZIZ,^∗∗Université de Tunis El Manar JEAN-FRANÇOIS DELMAS,^∗∗∗Ecole des Ponts

Abstract

We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton–Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton–Watson tree conditioned on having a large number of protected nodes.

Keywords:Galton–Watson tree; random tree; local limit; protected node 2010 Mathematics Subject Classification: Primary 60J80; 60B10

Secondary 05C05

1. Introduction

In [6] Kesten proved that a critical or subcritical Galton–Watson (GW) tree conditioned on reaching at least heighth converges in distribution (for the local topology on trees) ash goes to∞ toward the so-called sized-biased tree (that we call here Kesten’s tree and whose distribution is described in Section 3.2). Since then, other conditionings have been considered, see [1], [2], [4], and the references therein for recent developments on the subject.

A protected node is a node that is not a leaf and none of its offspring is a leaf. Precise asymptotics for the number of protected nodes in a conditioned GW tree have already been obtained in [3], [5], for instance. LetA(t)be the number of protected nodes in the treet. We remark that this functionalAis clearly monotone in the sense of [4] (using, for instance, (5.1));

therefore, using Theorem 2.1 of [4], we immediately find that a critical GW treeτ conditioned on{A(τ ) > n}converges in distribution toward Kesten’s tree asngoes to∞. Conditioning on {A(τ )=n}needs extra work and is the main objective of this paper. Using the general result of [1], if we have the following limit

n→+∞lim

P(A(τ )=n+1)

P(A(τ )=n) =1, (1.1) then the critical GW treeτ conditioned on{A(τ ) =n}converges in distribution also toward Kesten’s tree; see Theorem 5.1.

Received 10 September 2015; revision received 22 June 2016.

∗Postal address: Laboratoire MAPMO, Université d’Orléans, BP 6759, 45067 Orléans Cedex 2, France.

Email address: romain.abraham@univ-orleans.fr

∗∗Postal address: Institut préparatoire aux études scientifiques et techniques, Université de Tunis El Manar, 2070 La Marsa, Tunis, Tunisie.

∗∗∗Postal address: CERMICS, Ecole des Ponts, Université Paris-Est, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France.

In fact, the limit (1.1) can be seen as a special case of a more general problem: conditionally given the tree, we mark the nodes of the tree independently of the rest of the tree with a probability that depends only on the number of offspring of the nodes. Then we prove that a critical GW tree conditioned on the total number of marked nodes being large converges in distribution toward Kesten’s tree; see Theorem 3.1.

The paper is then organised as follows. We first recall briefly the framework of discrete trees, then we consider in Section 3 the problem of a marked GW tree and the proofs of the results are given in Section 4. In particular, in Lemma 4.1 we prove the limit (1.1) whenAis the number of marked nodes, and we deduce the convergence of a critical GW tree conditioned on the number of marked nodes toward Kesten’s tree in Theorem 3.1. Finally, in Section 5 we explain how the problem of protected nodes can be viewed as a problem on marked nodes.

and deduce the convergence in distribution of a critical GW tree conditioned on the number of protected nodes toward Kesten’s tree in Theorem 5.1.

2. Technical background on GW trees 2.1. The set of discrete trees

We denote byN= {0,1,2, . . .}the set of nonnegative integers and byN^∗= {1,2, . . .}the set of positive integers.

IfEis a subset ofN^∗, we call the span ofEthe greatest common divisor ofE. IfXis an integer-valued random variable, we call the span ofXthe span of{n >0;P(X=n) >0}.

We recall Neveu’s formalism [7] for ordered rooted trees. LetU=

n≥0(N^∗)ⁿbe the set of finite sequences of positive integers with the convention(N^∗)⁰= {∅}. Foru∈U, its length or generation|u| ∈Nis defined byu∈(N^∗)^|u|. Ifuandvare two sequences ofU, we denote byuvthe concatenation of the two sequences, with the convention thatuv=uifv =∅and uv=vifu=∅. The set of ancestors ofuis the set

An(u)= {v∈U; there existsw∈Usuch thatu=vw}.

Note thatubelongs to An(u). For two distinct elementsuandvofU, we denote byu < vthe lexicographic order onU, i.e.u < vifu∈An(v)andu=vor ifu=wiuandv=wj vfor somei, j ∈N^∗withi < j. We writeu≤vifu=voru < v.

A treetis a subset ofUthat satisfies the following.

• ∅∈t.

• Ifu∈tthen An(u)⊂t.

• For everyu∈t, there existsku(t)∈Nsuch that, for everyi∈N^∗,ui∈tif and only if 1≤i≤ku(t).

The vertex∅is called the root oft. The integerk_u(t)represents the number of offspring of the vertexu∈t. The set of children of a vertexu∈tis given by

C_u(t)= {ui; 1≤i≤k_u(t)}.

By convention, we setk_u(t)= −1 ifu∈t.

A vertexu ∈t is called a leaf ifk_u(t)=0. We denote byL0(t)the set of leaves oft. A vertexu∈tis called aprotected nodeifC_u(t)=∅andC_u(t)∩L0(t)=∅, that is,uis not

a leaf and none of its children is a leaf. Foru ∈ t, we defineF_u(t), the fringe subtree oft aboveu, as

F_u(t)= {v∈t; u∈An(v)} = {uv;v ∈S_u(t)}

withS_u(t)= {v∈U;uv∈t}.

Note thatS_u(t)is a tree. We denote byTthe set of trees and byT0= {t∈T; card(t) <+∞}

the subset of finite trees.

We say that a sequence of trees (t_n, n ∈ N)converges locally to a tree t if and only if lim_n→∞k_u(t_n)=k_u(t)for allu∈U. Let(T_n, n∈N)andT beT-valued random variables.

We denote by dist(T )the distribution of the random variableT and write

n→+∞lim dist(T_n)=dist(T )

for the convergence in distribution of the sequence(T_n, n∈N)toT with respect to the local topology.

Ift,t∈Tandx∈L0(t), we denote by

txt= {u∈t} ∪ {xv;v∈t}

the tree obtained by grafting the treeton the leafx of the treet. For everyt ∈Tand every x∈L0(t), we shall consider the set of trees obtained by grafting a tree on the leafx oft, i.e.

T(t, x)= {t_xt; t∈T}.

2.2. GW trees

Letp=(p(n), n∈N)be a probability distribution onN. We assume that p(0) >0, p(0)+p(1) <1, and μ:=^+∞

n=0

np(n) <+∞. (2.1) AT-valued random variableτ is a GW tree with offspring distributionpif the distribution ofk∅(τ )ispand it enjoys the branching property: forn∈N^∗, conditionally on{k∅(τ )=n}, the subtrees(S1(τ ), . . . , Sn(τ ))are independent and distributed as the original treeτ.

The GW tree and the offspring distribution are called critical (respectively subcritical, super- critical) ifμ=1 (respectivelyμ <1, μ >1).

3. Conditioning on the number of marked vertices 3.1. Definition of the marking procedure

We begin with a fixed tree t. We add marks on the vertices oft in an independent way such that the probability of adding a mark on a nodeudepends only on the number of children ofu. More precisely, we consider a mark functionq:N→ [0,1]and a family of independent Bernoulli random variables(Z_u(t), u∈t)such that, for allu∈t,

P(Zu(t)=1)=1−P(Zu(t)=0)=q(ku(t)).

The vertexuis said to have a mark ifZ_u(t)=1. We denote byM(t)= {u∈t;Z_u(t)=1}

the set of marked vertices and byM(t)its cardinal. We call(t,M(t))a marked tree.

A marked GW tree with offspring distributionpand mark functionqis a couple(τ,M(τ )), withτ a GW tree with offspring distributionpand conditionally on{τ =t}the set of marked verticesM(τ )is distributed asM(t).

Remark 3.1. Note that, forA⊆N, if we setq(k)=1_{k∈A}then the setM(t)is just the set of vertices with out-degree (i.e. number of offspring) inAconsidered in [1], [8]. Hence, the above construction can be seen as an extension of this case.

3.2. Kesten’s tree

Letpbe an offspring distribution satisfying assumption (2.1) withμ≤1 (i.e. the associated GW process is critical or subcritical). We denote byp^∗ =(p^∗(n) = np(n)/μ, n ∈ N)the corresponding size-biased distribution.

We define an infinite random treeτ^∗(the size-biased tree that we call Kesten’s tree in this paper) whose distribution is described as follows.

There exists a unique infinite sequence(v_k, k∈N^∗)of positive integers such that, for every h∈N, v1· · ·v_h∈τ^∗, with the convention thatv1· · ·v_h=∅ifh=0. The joint distribution of(v_k, k ∈ N^∗)andτ^∗is determined recursively as follows. For eachh∈ N, conditionally given(v1, . . . , vh)and{u∈τ^∗; |u| ≤h}the treeτ^∗up to levelh, we have the following.

• The number of children (k_u(τ^∗), u ∈ τ^∗,|u| = h) are independent and distributed according topifu=v1· · ·v_hand according top^∗ifu=v1. . . v_h.

• Given{u∈τ^∗; |u| ≤h+1}and(v1, . . . , v_h), the integerv_h+1is uniformly distributed on the set of integers{1, . . . , k_v1···vh(τ^∗)}.

Remark 3.2. Note that by construction, almost surelyτ^∗ has a unique infinite spine. And following Kesten [6], the random treeτ^∗ can be viewed as the treeτ conditioned on non- extinction.

Fort∈T0andx ∈L0(t), we have

P(τ^∗∈T(t, x))= P(τ =t) μ^|x|p(0).

3.3. Main theorem

Theorem 3.1. Letp be a critical offspring distribution that satisfies assumption (2.1). Let (τ,M(τ ))be a marked GW tree with offspring distributionpand mark functionq such that p(k)q(k) >0 for somek ∈ N. For everyn ∈ N^∗, letτ_nbe a tree whose distribution is the conditional distribution ofτ given{M(τ )=n}. Letτ^∗be a Kesten’s tree associated withp.

Then we have

n→+∞lim dist(τ_n)=dist(τ^∗),

where the limit has to be understood along a subsequence for whichP(M(τ )=n) >0.

Remark 3.3. If, for everyk ∈ N, 0 < q(k) <1 thenP(M(τ )) =n) >0 for everyn ∈ N, hence, the above conditioning is always valid.

4. Proof of Theorem 3.1

Setγ = P(M(τ ) > 0). Since there existsk ∈ Nwithp(k)q(k) > 0, we haveγ > 0.

A sufficient condition (but not necessary) to haveP(M(τ )=n) >0 for every large enoughn is to assume thatγ <1 (see Lemma 4.2 and Section 4.4). Takingq =1_A, see Remark 3.1 for 0∈A⊂Nimplies thatγ =1 and some periodicity may occur.

The following result is the analogue in the random case of Theorem 3.1 in [1] and its proof is in fact a straightforward adaptation of the proof in [1] by using the following.

(i) M(t)≤card(t).

(ii) For everyt ∈ T0,x ∈ L0(t), and t ∈ T, it follows thatM(t_xt)is distributed asM(tˆ )+M(t)−1_{Zx(t)=1}, whereM(tˆ )is distributed asM(t)and is independent ofM(t).

Proposition 4.1. Let n0 ∈ N∪ {∞}. Assume that P(M(τ ) ∈ [n, n+n0)) > 0 for large enoughn. Then, if

n→+∞lim

P(M(τ )∈ [n+1, n+1+n0)

P(M(τ )∈ [n, n+n0)) =1, (4.1) we have

n→+∞lim dist(τ | M(τ )∈ [n, n+n0))=dist(τ^∗).

Proof. According to Lemma 2.1 in [1], a sequence(T_n, n ∈ N)of finite random trees converges in distribution (with respect to the local topology) to some Kesten’s treeτ^∗if and only if, for every finite treet∈T0and every leafx∈L0(t),

n→+∞lim P((T_n∈T(t, x))=P(τ^∗∈T(t, x)) and lim

n→+∞P(T_n=t)=0. (4.2) Lett ∈ T0andx ∈ L0(t). We setD(t, x) = M(t)−1_{Zx(t)=1}. Note thatD(t, x) ≤ card(t)−1. Elementary computations yield, for everyt∈T0,

P(τ =t_xt)= 1

p(0)P(τ =t)P(τ =t) and P(τ^∗∈T(t, x))= 1

p(0)P(τ =t).

Asτ is almost surely finite, we have

P(τ ∈T(t, x), M(τ )∈ [n, n+n0))

=

t∈T0

P(τ =txt, M(τ )∈ [n, n+n0))

=

t∈T0

P(τ =txt)P(M(txt)∈ [n, n+n0))

=

t∈T0

P(τ =t)P(τ =t)

p(0) P(M(tˆ )+D(t, x)∈ [n, n+n0))

=P(τ^∗∈T(t, x))P(M(τ )ˆ +D(t, x)∈ [n, n+n0)).

Note that

P(M(τ )ˆ +D(t, x)∈ [n, n+n0))

=

card(t)−1

k=0

P(M(τ )ˆ +D(t, x)∈ [n, n+n0)|D(t, x)=k)P(D(t, x)=k)

=

card(t)−1

k=0

P(M(τ )∈ [n−k, n+n0−k))P(D(t, x)=k).

Then, using assumption (4.1), we obtain

n→+∞lim

P(M(τ )ˆ +D(t, x)∈ [n, n+n0)) P(M(τ )∈ [n, n+n0)) =1, that is,

n→+∞lim P(τ ∈T(t, x)| M(τ )∈ [n, n+n0))=P(τ^∗∈T(t, x)).

This proves the first limit of (4.2).

The second limit is immediate since, for everyn≥card(t),

P(τ =t | M(τ )∈ [n, n+n0))=0.

The main ingredient for the proof of Theorem 3.1 is the following lemma.

Lemma 4.1. Letd be the span of the random variableM(τ )−1. We have

n→+∞lim

P(M(τ )∈ [n+1, n+1+d))

P(M(τ )∈ [n, n+d)) =1. (4.3) The end of this section is devoted to the proof of Lemma 4.1, see Section 4.4, which follows the ideas of the proof of Theorem 5.1 of [1].

4.1. Transformation of a subset of a tree onto a tree

We recall Rizzolo’s map [8] which, from t ∈ T0 and a nonempty subset Aof t, builds a treet_A such that card(A) = card(t_A). We will give a recursive construction of this map φ:(t, A)→t_A =φ(t, A). We will check in the next section that this map is such that ifτ is a GW tree thenτ_Awill also be a GW tree for a well chosen subsetAofτ. In Figure 1 we show an example of a treet, a setA, and the associated treet_Awhich helps to understand the construction.

For a vertexu∈t, recall thatC_u(t)is the set of children ofuint. We define, foru∈t, R_u(t)=

w∈An(u)

{v∈C_w(t);u < v}

the vertices oftwhich are larger thanufor the lexicographic order and are children ofuor of one of its ancestors. For a vertexu∈ t, we shall considerAuthe set of elements ofAin the fringe subtree aboveu, i.e.

Au=A∩Fu(t)=A∩ {uv; v∈Su(t)}. (4.4)

1

2 3

4 5

6 7

1 2 3

4 5 6

7 1

2 3

4 5

6 7

Figure1:Left:a treetand the setA.Centre:the fringe subtrees rooted at the vertices inRu0(t). Right: the treet_A. The labels have no signification, they only show which

node oftcorresponds to a node oft_A.

Lett∈T0andA⊂tsuch thatA=∅. We shall definet_A=φ(t, A)recursively. Letu0be the smallest (for the lexicographic order) element ofA. Consider the fringe subtrees oftthat are rooted at the vertices inR_u0(t)and contain at least one vertex inA, that is(F_u(t);u∈R^A_u0(t)), with

R_u^A₀(t)= {u∈Ru0(t);Au =∅} = {u∈Ru0(t); there existsv∈Asuch thatu∈An(v)}.

Define the number of children of the root of treet_Aas the number of those fringe subtrees k_∅(t_A)=card(R_u^A₀(t)).

Ifk∅(tA)=0 settA = {∅}. Otherwise, letu1 <· · · < uk∅(tA)be the ordered elements of R_u^A₀(t)with respect to the lexicographic order onU. And we definetA=φ(t, A)recursively by

Fi(tA)=φ(Fui(t), Aui) for 1≤i≤k∅(tA). (4.5) Since card(Aui) < card(A), we deduce thattA = φ(t, A)is well defined and is a tree by construction. Furthermore, we clearly have thatAandtAhave the same cardinal, i.e.

card(t_A)=card(A). (4.6)

4.2. Distribution of the number of marked nodes

Let(τ,M(τ ))be a marked GW tree with critical offspring distributionpsatisfying (2.1) and mark functionq. Recall thatγ =P(M(τ ) >0)=P(M(τ )=∅).

Let((Xi, Zi), i ∈ N^∗)be independent and identically distributed random variables such thatXi is distributed according topandZi is conditionally onXi Bernoulli with parameter q(Xi). We have the following definitions.

• G=inf{k∈N^∗; _k

i=1(X_i−1)= −1}.

• N =inf{k∈N^∗;Z_k=1}.

• ˜Xa random variable distributed as 1+_N

i=1(X_i −1)conditionally on{N ≤G}.

• Y a random variable which is conditionally onX˜ binomial with parameter(X, γ ).˜ We say that a probability distribution onNis aperiodic if the span of its support restricted toN^∗is 1. The following result is immediate as the distributionpofX1satisfies (2.1).

Lemma 4.2. The distribution ofY satisfies (2.1) and ifγ <1then it is aperiodic.

Recall that for a treet∈T0, we have

u∈t

(k_u(t)−1)= −1 (4.7)

and

u∈t,u<v(k_u(t)−1) > −1 for anyv ∈ t. We deduce thatGis distributed according to card(τ )and, thus,Nis distributed like the index of the first marked vertex along the depth-first walk ofτ. Then, we have

γ =P(N≤G). (4.8)

We denote by(τ⁰,M(τ⁰))a random marked tree distributed as(τ,M(τ ))conditioned on {M(τ )=∅}. By construction, card(τ⁰)is distributed asGconditioned on{N ≤G}. Lemma 4.3. Under the hypothesis of this section, it holds thatτ_M(τ⁰ 0) =φ(τ⁰,M(τ⁰))is a critical GW tree with the law ofY as offspring distribution.

4.3. Proof of Lemma 4.3

In order to simplify notation, we writeτ˜ forτ_M(τ⁰ 0)=φ(τ⁰,M(τ⁰))and, foru ∈ τ⁰, we setR_uforR_u(τ⁰).

Lemma 4.4. The random treeτ˜is a GW tree with offspring distribution the law ofY. Proof. Letu0be the smallest (for the lexicographic order) element ofM(τ⁰). The branching property of GW trees implies that, conditionally givenu0andRu0, the fringe subtrees ofτ⁰ rooted at the vertices inRu0,(Su(τ⁰), u∈Ru0)are independent and distributed asτ. Recall notation (4.4) so that the set of marked vertices of the fringe subtree rooted atuisMu(τ⁰)= M(τ⁰)∩Fu(τ⁰). DefineM˜u(τ⁰)= {v;uv ∈ Mu(τ⁰)}the corresponding marked vertices ofS_u(t). Then, the construction of the marksM(τ ) implies that the corresponding marked trees((S_u(τ⁰),M˜_u(τ⁰)), u∈ R_u0)are independent and distributed as(τ,M(τ )). Note that, foru∈R_u0, the fringe subtreeF_u(τ⁰)contains at least one mark if and only ifubelongs to

R_u^M(τ₀ ⁰⁾= {u∈R_u0;there existsv∈M(τ⁰)such thatu∈An(v)}.

Then by considering only the fringe subtrees containing at least one mark, we find that, conditionally onR^M(τ_u0 ⁰⁾, the subtrees((S_u(τ⁰),M˜_u(τ⁰)), u ∈ R^M(τ_u0 ⁰⁾)are independent and distributed as(τ⁰,M(τ⁰)). We deduce from the recursive construction of the mapφ, see (4.5), thatτ˜is a GW tree. Note that the offspring distribution ofτ˜is given by the distribution of the cardinal ofR^M(τ_u0 ⁰⁾. We now compute the corresponding offspring distribution. We first give an elementary formula for the cardinal ofR_u(t). Lett ∈ T0andu ∈ t. Consider the tree t=R_u(t)∪ {v∈t;v≤u}. Using (4.7) fort, we obtain

−1=

v∈t

(k_v(t)−1)=

v∈t;v≤u

(k_v(t)−1)+

v∈Ru(t)

(−1).

From this we obtain card(R_u(t)) = 1+

v∈t;v≤u(k_v(t)−1). We deduce from the defi- nition ofX˜ that card(R_u0)is distributed as X. We deduce from the first part of the proof˜ that conditionally on card(R_u0), the distribution of card(R^M(τ_u0 ⁰⁾)is binomial with parameter (card(R_u0(τ⁰)), γ ). It follows that the offspring distribution ofτ˜is given by the law ofY. Lemma 4.5. The GW treeτ˜is critical.

Proof. Since the offspring distribution is the law ofY, we need to check thatE[Y] =1 is γE[ ˜X] =1 sinceY is conditionally onX˜ binomial with parameter(X, γ )˜ .

Recall thatNhas finite expectation asP(Z1=1) >0, is not independent of(X_i)_i∈N^∗, and is a stopping time with respect to the filtration generated by((X_i, Z_i), i∈N^∗). Using Wald’s equality andE[X_i] =1, we obtainE[_N

i=1(X_i −1)] =0 and, thus, using the definition ofX˜ as well as (4.8),

γE[ ˜X] =γ +E _N

i=1

(X_i−1)1_{N≤G}

=γ−E _N

i=1

(X_i−1)1_{N>G}

.

We have E

_N

i=1

(X_i−1)1_{N>G}

=E _G

i=1

(X_i −1)1_{N>G}

+P(N > G)E _N

i=1

(X_i−1)

= −P(N > G)

=γ−1,

where we used the strong Markov property of((X_i, Z_i), i∈N^∗)at the stopping timeGfor the first equation, the definition ofT and Wald’s equality for the second, and (4.8) for the third.

We deduce thatE[Y] =γE[ ˜X] =1, which completes the proof.

4.4. Proof of (4.3)

According to Lemma 4.3 and (4.6), it follows thatM(τ⁰)is distributed as the total size of a critical GW whose offspring distribution satisfies (2.1). The proof of Proposition 4.3 of [1]

(see Equation (4.15) in [1]) entails that ifτis a critical GW tree, then, ifddenotes the span of the random variable card(τ)−1, we have

n→∞lim

P(card(τ)∈ [n+1, n+1+d))

P(card(τ)∈ [n, n+d)) =1.

5. Protected nodes

Recall that a node of a treetis protected if it is not a leaf and none of its offspring is a leaf.

We denote byA(t)the number of protected nodes of the treet.

Theorem 5.1. Letτ be a critical GW tree with offspring distributionp satisfying (2.1) and letτ^∗be the associated Kesten’s tree. Letτ_nbe a random tree distributed asτ conditionally given{A(τ )=n}. Then

n→+∞lim dist(τn)=dist(τ^∗).

Proof. Note thatP(A(τ )=n) >0 for alln∈ N. Note that the functionalAsatisfies the additive property of [1], namely, for everyt∈T, everyx ∈L0(t), and everyt∈Tthat is not reduced to the root, we have

A(t_xt)=A(t)+A(t)+D(t, x), (5.1) whereD(t, x) = 1 ifx is the only child of its first ancestor which is a leaf (therefore, this ancestor becomes a protected node in t_x t) and D(t, x) = 0 otherwise. According to Theorem 3.1 of [1], to complete the proof it is enough to check that

n→+∞lim

P(A(τ )=n+1)

P(A(τ )=n) =1. (5.2) For a treet = {∅}, lett_N^∗ =φ(t,t\L0(t))be the tree obtained fromt by removing the leaves. Letτ⁰be a random tree distributed asτconditioned to{k_∅(τ ) >0}. Using Theorem 6 and Corollary 2 of [8] withA=N^∗(or Lemma 4.3 withq(k)=1_{k>0}), it follows thatτ_N⁰∗is a critical GW tree with offspring distribution

p_N^∗(k)= ^+∞

n=max(k,1)

p(n) n

k (p(0))^n−k(1−p(0))^k−1, k∈N.

Figure2: The treesτ⁰,τ_N⁰∗, andτˆ.

Conditionally given {τ_N⁰∗ = t}, we consider independent random variables(W (u), u ∈ t) taking values inN^∗whose distributions are given, for allu ∈ t, byP(W (u) = 0) = 0 for k_u(t)=0 and otherwise, fork_u(t)+n >0 (remark thatp_N^∗(k_u(t)) >0), by

P(W (u)=n)=p(ku(t)+n) p_N^∗(k_u(t))

ku(t)+n

n p(0)ⁿ(1−p(0))^ku(t)−1. In particular, forku(t) >0, we have

P(W (u)=0)= p(k_u(t))

p_N^∗(k_u(t))(1−p(0))^ku(t)−1. (5.3) Then, we define a new treeτˆby grafting, on every vertexuofτ_N⁰∗,W (u)leaves in a uniform manner; see Figure 2.

More precisely, givenτ_N⁰∗ and(W (u), u ∈ τ_N⁰∗), we define a treeτˆ and a random map ψ: τ_N⁰∗ −→ ˆτrecursively in the following way. We setψ(∅)=∅. Then, givenk_∅(τ_N⁰∗)=k, we setk∅(τ )ˆ =k+W (∅). We also consider a family(i1, . . . , ik)of integer-valued random variables such that(i1, i2−i1, . . . , ik−ik−1, W (u)+k+1−ik)is a uniform positive partition ofW (u)+k+1. Then, for everyj ≤ksuch thatj ∈ {i1, . . . , ik}, we setkj(ˆτ )=0, i.e. these are leaves ofτˆ. For every 1≤j ≤k, we setψ(j )=ijand apply to them the same construction

as for the root and so on.

Lemma 5.1. The new treeτˆis distributed as the original treeτ⁰.

Proof. Lett ∈ T0. AsP(ˆτ = {∅})=0, we assume thatk_∅(t) >0. Lett_N^∗ be the tree obtained fromtby removing the leaves. Using (4.7), we have

P(τˆ=t)=

u∈t_N∗

p_N^∗(k_u(t_N^∗))P(W (u)=k_u(t)−k_u(t_N^∗))

k_u(t) k_u(t)−k_u(t_N^∗)

−1

= P(τ =t) 1−p(0)

=P(τ⁰=t).

Note that the protected nodes ofτˆare exactly the nodes ofτ_N⁰∗on which we did not add leaves, i.e. for whichW (u)=0. If we setM(τ_N⁰∗)= {u∈τ_N⁰∗, W (u)=0}, we haveM(τ_N⁰∗)=A(τ ).ˆ

Using (5.3), we find that the corresponding mark functionqis given by q(k)= p(k)(1−p(0))^k−1

p_N^∗(k) 1_{k≥1}. Asτˆis distributed asτ⁰, we have

n→+∞lim

P(A(τ⁰)=n+1)

P(A(τ⁰)=n) = lim

n→+∞

P(A(τ )ˆ =n+1) P(A(τ )ˆ =n) = lim

n→+∞

P(M(τ_N⁰∗)=n+1) P(M(τ_N⁰∗)=n) .

Asτ_N⁰∗is a critical GW tree, from Lemma 4.1 we deduce that

n→+∞lim

P(M(τ_N⁰∗)=n+1) P(M(τ_N⁰∗)=n) =1.

AsP(A(τ ) = n) =P(A(τ ) = n|k∅(τ ) > 0)P(k∅(τ ) > 0)andP(A(τ ) = n|k∅(τ ) >

0)=P(A(τ⁰)=n)forn≥2, we obtain (5.2) and, hence, complete the proof.

Acknowledgement

We would like to thank the anonymous referee for his/her useful comments which helped to improve the paper.

References

[1] Abraham, R. and Delmas, J.-F.(2014). Local limits of conditioned Galton–Watson trees: the infinite spine case.

Electron. J. Prob.19,19 pp.

[2] Abraham, R. and Delmas, J.-F.(2015). An introduction to Galton–Watson trees and their local limits. Lecture given in Hammamet, December 2014. Available at https://arxiv.org/abs/1506.05571.

[3] Devroye, L. and Janson, S.(2014). Protected nodes and fringe subtrees in some random trees.Electron.

Commun. Prob.19,10 pp.

[4] He, X.(2015). Local convergence of critical random trees and continuous-state branching processes. Preprint Available at https://arxiv.org/abs/1503.00951.

[5] Janson, S.(2016). Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees.Random Structures Algorithms48,57–101.

[6] Kesten, H.(1986). Subdiffusive behavior of random walk on a random cluster.Ann. Inst. H. Poincaré Prob.

Statist.22,425–487.

[7] Neveu, J.(1986). Arbres et processus de Galton–Watson.Ann. Inst. H. Poincaré Prob. Statist.22,199–207.

[8] Rizzolo, D.(2015). Scaling limits of Markov branching trees and Galton–Watson trees conditioned on the number of vertices with out-degree in a given set.Ann. Inst. H. Poincaré Prob. Statist.51,512–532.