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LOCAL LIMITS OF GALTON-WATSON TREES
CONDITIONED ON LARGE WIDTH

Aymen Bouaziz

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Aymen Bouaziz. LOCAL LIMITS OF GALTON-WATSON TREES CONDITIONED ON LARGE WIDTH. 2023. hal-04101663

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Preprint submitted on 21 May 2023

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LOCAL LIMITS OF GALTON-WATSON TREES
CONDITIONED ON LARGE WIDTH

AYMEN BOUAZIZ

ABSTRACT. We study the local convergence of critical Galton-Watson trees under various conditionings. We give a sufficient condition, which serves to cover all the previous cases, for the convergence in distribution of a conditioned Galton-Watson tree to Kesten's tree. We also propose an other proof to give the limit in distribution of a critical Galton-Watson tree, with bounded support, conditioned on having a large width.

# 1. Introduction

In [7], Kesten proved that a critical or sub-critical Galton-Watson (GW) tree conditioned on reaching at least height $h$ converges in distribution (for the local topology on trees) as $h$ goes to infinity toward the so-called sized-biased tree (that we call here Kesten's tree and whose distribution is described in Section 2.3). Since then, several different conditioning have been studied, in particular, the conditioning of extinction after large time, the conditioning of large total population size and the conditioning of large number of leaves. In [2], Abraham and Delmas provided a criterion for local convergence of finite random trees to Kestens tree, then gave short and elementary proofs of essentially all previous related results and some new ones. Later, in [1], we prove that a critical Galton-Watson tree conditioned on having a large number of marked vertices converges in distribution to Kesten's tree and 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.

Let $L(t)$ be the width of the tree $t$. Remark that this functional $L$ is clearly monotone in the sense of [5]; therefore, using Theorem 2.1 of [5], we immediately get that a critical GW tree $	au$ conditioned on $\{L(\tau) > n\}$ converges in distribution toward Kesten's tree as $n$ goes to infinity. In this paper, we propose another proof by generalizing somewhat the monotonicity property. Notice that the functional $L$ does not satisfy additivity in [2]. Thus, considering the conditioning event $\{L(\tau) = n\}$ is still an open problem.

<footnote>Date: May 12, 2023.</footnote>

<footnote n="2010">2010 Mathematics Subject Classification. 60J80, 60B10, 05C05.</footnote>

<footnote>Key words and phrases. Galton-Watson tree; random trees; local limits; width.</footnote>
We say that a probability distribution $p = (p(0), p(1), \dots)$ on non-negative integers is bounded, if the set $\{n \in \mathbb{N}, p(n) > 0\}$ is bounded. For any critical and bounded distribution $p$ satisfying assumption (1), Xin He [5] provides a positive answer to this question, more precisely he proves that a critical GW tree $\tau$ conditioned on $\{L(\tau) = n\}$ converges in distribution toward Kesten's tree as $n$ goes to infinity.

Provide an other proof of this result is the main objective of this paper. On the technical level, our proofs are extremely short and elementary, thanks in particular to the convenient framework in [2].

The paper is then organized as follows: In section 2, we recall briefly the framework we use for discrete trees and define the Galton-Watson tree $\tau$ and Kesten's tree $\tau^*$ associated with offspring distribution $p$. In section 3, we state and prove our general result on local convergence of conditioned critical and sub-critical Galton-Watson trees and we apply it to the conditioning of large width, in the critical case, in Corollary 3.6, which is one of our main objective of this paper.

In section 4, we study the conditioning on having large width of a critical Galton-Watson tree, with bounded support, where we give an elementary and short proof. Finally, we generalized this result by considering a type of conditioning never treated.

## 2. Technical background on GW trees

### 2.1. The set of discrete trees.

We denote by $\mathbb{N} = \{0, 1, 2, \dots\}$ the set of non-negative integers and by $\mathbb{N}^* = \{1, 2, \dots\}$ the set of positive integers. We recall Neveu's formalism [8] for ordered rooted trees. Let

$$ \mathcal{U} = \bigcup_{n \ge 0} (\mathbb{N}^*)^n $$

be the set of finite sequences of positive integers with the convention $(\mathbb{N}^*)^0 = \{\emptyset\}$. For $u \in \mathcal{U}$, its length or generation $|u| \in \mathbb{N}$ is defined by $u \in (\mathbb{N}^*)^{|u|}$. If $u$ and $v$ are two sequences of $\mathcal{U}$, we denote by $uv$ the concatenation of the two sequences, with the convention that $uv = u$ if $v = \emptyset$ and $uv = v$ if $u = \emptyset$. The set of ancestors of $u$ is the set

$$ \mathrm{An}(u) = \{v \in \mathcal{U}; \exists w \in \mathcal{U} \text{ such that } u = vw\}. $$

The most recent common ancestor of a subset $s$ of $\mathcal{U}$, denoted by $M(s)$, is the unique element $u$ of $\cap_{u \in s} \mathrm{An}(u)$ with maximal length $|u|$. For two distinct elements $u$ and $v$ of $\mathcal{U}$, we denote by $u < v$ the lexicographic order on $\mathcal{U}$ i.e. $u < v$ if $u \in \mathrm{An}(v)$ and $u \neq v$ or if $u = wiu'$ and $v = wjv'$ for some $i, j \in \mathbb{N}^*$ with $i < j$. We write $u \le v$ if $u = v$ or $u < v$.

A tree $t$ is a subset of $\mathcal{U}$ that satisfies:

$$ \bullet \emptyset \in t. $$
• If $u \in \mathbf{t}$, then $\mathrm{An}(u) \subset \mathbf{t}$.

• For every $u \in \mathbf{t}$, there exists $k_u(\mathbf{t}) \in \mathbb{N}$ such that, for every $i \in \mathbb{N}^*$, $u_i \in \mathbf{t}$ iff $1 \le i \le k_u(\mathbf{t})$.

The vertex $\emptyset$ is called the root of $\mathbf{t}$. The integer $k_u(\mathbf{t})$ represents the number of offsprings of the vertex $u \in \mathbf{t}$, and we call it the outdegree of the node $u$ in the tree $\mathbf{t}$. The maximal outdegree $M(\mathbf{t})$ of a tree $\mathbf{t}$ is defined by

$$M(\mathbf{t}) = \sup\{k_u(\mathbf{t}), u \in \mathbf{t}\}$$

The set of children of a vertex $u \in \mathbf{t}$ is given by:

$$C_u(\mathbf{t}) = \{u_i; 1 \le i \le k_u(\mathbf{t})\}.$$

By convention, we set $k_u(\mathbf{t}) = -1$ if $u \notin \mathbf{t}$.

A vertex $u \in \mathbf{t}$ is called a leaf if $k_u(\mathbf{t}) = 0$. We denote by $\mathcal{L}_0(\mathbf{t})$ the set of leaves of $\mathbf{t}$. A vertex $u \in \mathbf{t}$ is called a protected node if $C_u(\mathbf{t}) \ne \emptyset$ and $C_u(\mathbf{t}) \cap \mathcal{L}_0(\mathbf{t}) = \emptyset$, that is $u$ is not a leaf and none of its children is a leaf.

For a tree $\mathbf{t}$, we denote by

$$\mathcal{Z}(\mathbf{t}) = \mathrm{Card}(\{u \in \mathbf{t}, |u| = n\})$$

the size of the $n$-th generation of $\mathbf{t}$, the height of $\mathbf{t}$ is defined by

$$H(\mathbf{t}) = \sup\{|u|, u \in \mathbf{t}\}$$

and can be infinite. We define the width $L(\mathbf{t})$ of $\mathbf{t}$ as:

$$L(\mathbf{t}) = \sup_{k \ge 0} \mathcal{Z}_k(\mathbf{t}).$$

We denote by $\mathbb{T}$ the set of trees, by

$$\mathbb{T}_0 = \{\mathbf{t} \in \mathbb{T}; \mathrm{Card}(\mathbf{t}) < +\infty\}$$

the subset of finite trees and by

$$\mathbb{T}_1 = \{\mathbf{t} \in \mathbb{T}; \lim_{n \to +\infty} |M(\{u \in \mathbf{t}; |u| = n\})| = +\infty\}$$

the subset of trees with a unique infinite spine.

We say that a sequence of trees $(\mathbf{t}_n, n \in \mathbb{N})$ converges locally to a tree $\mathbf{t}$ if and only if $\lim_{n \to +\infty} k_u(\mathbf{t}_n) = k_u(\mathbf{t})$ for all $u \in \mathcal{U}$.

Let $(T_n, n \in \mathbb{N})$ and $T$ be $\mathbb{T}$-valued random variables. We denote by $\mathrm{dist}(T)$ the distribution of the random variable $T$ and write

$$\lim_{n \to +\infty} \mathrm{dist}(T_n) = \mathrm{dist}(T)$$

for the convergence in distribution of the sequence $(T_n, n \in \mathbb{N})$ to $T$ with respect to the local topology. If $\mathbf{t}, \mathbf{t}' \in \mathbb{T}$ and $x \in \mathcal{L}_0(\mathbf{t})$ we denote by

$$\mathbf{t} \otimes_x \mathbf{t}' = \{u \in \mathbf{t}\} \cup \{xv; v \in \mathbf{t}'\}$$
the tree obtained by grafting the tree $t'$ on the leaf $x$ of the tree $t$. For every $t \in \mathbb{T}$ and every $x \in \mathcal{L}_0(t)$, we shall consider the set of trees obtained by grafting a tree on the leaf $x$ of $t$:

$$ \mathbb{T}(t, x) = \{t \circledast_x t'; t' \in \mathbb{T}\}. $$

For convergence in distribution in the set $\mathbb{T}_0 \cup \mathbb{T}_1$, we recall the following key characterization in [3].

**Lemma 2.1.** Let $(T_n)_{n \in \mathbb{N}}$ and $T$ be random trees taking values in the set $\mathbb{T}_0 \cup \mathbb{T}_1$. Then the sequence $(T_n)_{n \in \mathbb{N}}$ converges in distribution to $T$ if and only if:

(1) for every finite tree $t \in \mathbb{T}_0$, $\lim_{n \to +\infty} \mathbb{P}(T_n = t) = \mathbb{P}(T = t)$;

(2) for every finite tree $t \in \mathbb{T}_0$ and every leaf $x$ of $t$,

$$ \liminf_{n \to +\infty} \mathbb{P}(T_n \in \mathbb{T}(t, x)) \geq \mathbb{P}(T \in \mathbb{T}(t, x)). $$

## 2.2. Galton Watson trees.

Let $p = (p(n), n \in \mathbb{N})$ be a probability distribution on $\mathbb{N}$. We assume that

$$ (1) \quad p(0) > 0, p(0) + p(1) < 1, \text{ and } \mu := \sum_{n=0}^{+\infty} np(n) < +\infty. $$

A $\mathbb{T}$-valued random variable $\tau$ is a GW tree with offspring distribution $p$ if the distribution of $k_0(\tau)$ is $p$ and it enjoys the branching property: for $n \in \mathbb{N}^*$, conditionally on $\{k_0(\tau) = n\}$, the subtrees $(S_1(\tau), \dots, S_n(\tau))$ are independent and distributed as the original tree $\tau$.

The GW tree and the offspring distribution are called critical (resp. sub-critical, super-critical) if $\mu = 1$ (resp. $\mu < 1$, $\mu > 1$).

In the critical and sub-critical case, we have that a.s $\tau$ belongs to $\mathbb{T}_0$.

## 2.3. Kesten's tree.

Let $p$ be an offspring distribution satisfying Assumption (1) with $\mu \le 1$ (i.e. the associated GW process is critical or sub-critical). We denote by $p^* = (p^*(n) = np(n)/\mu, n \in \mathbb{N})$ the corresponding size-biased distribution.

We define an infinite random tree $\tau^*$ (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 $(v_k, k \in \mathbb{N}^*)$ of positive integers such that, for every $h \in \mathbb{N}$, $v_1 \cdots v_h \in \tau^*$, with the convention that $v_1 \cdots v_h = \emptyset$ if $h = 0$. The joint distribution of $(v_k, k \in \mathbb{N}^*)$ and $\tau^*$ is determined recursively as follows. For each $h \in \mathbb{N}$, conditionally given $(v_1, \dots, v_h)$ and $\{u \in \tau^*; |u| \le h\}$ the tree $\tau^*$ up to level $h$, we have:
• The number of children ($k_u(\tau^*)$, $u \in \tau^*$, $|u| = h$) are independent and distributed according to $p$ if $u \neq v_1 \cdots v_h$ and according to $p^*$ if $u = v_1 \cdots v_h$.

• Given $\{u \in \tau^*; |u| \le h+1\}$ and $(v_1, \dots, v_h)$, the integer $v_{h+1}$ is uniformly distributed on the set of integers $\{1, \dots, k_{v_1 \cdots v_h}(\tau^*)\}$.

**Remark 2.2.** Notice that by construction, a.s. $\tau^* \in \mathbb{T}_1$ has a unique infinite spine. And following Kesten [7], the random tree $\tau^*$ can be viewed as the tree $\tau$ conditioned on non extinction.

Following [3], for $\mathbf{t} \in \mathbb{T}_0$ and $x \in \mathcal{L}_0(\mathbf{t})$, we have:

$$ (2) \qquad \mathbb{P}(\tau^* \in \mathbb{T}(t, x)) = \frac{\mathbb{P}(\tau = t)}{\mu^{|x|}p(0)} = \mu^{-|x|}\mathbb{P}(\tau \in \mathbb{T}(t, x)). $$

In the particular case of a critical offspring distribution ($\mu = 1$), we get for all $\mathbf{t} \in \mathbb{T}_0$ and $x \in \mathcal{L}_0(\mathbf{t})$:

$$ \mathbb{P}(\tau^* \in \mathbb{T}(t, x)) = \mathbb{P}(\tau \in \mathbb{T}(t, x)). $$

### 3. Main result

Let $A$ be an integer-valued function defined on $\mathbb{T}$ which is finite on $\mathbb{T}_0$ and let $n_0 \in \mathbb{N} \cup \{+\infty\}$ be given. We define for all $n \in \mathbb{N}^*$, the subset of trees

$$ A_n = \{\mathbf{t} \in \mathbb{T}; A(\mathbf{t}) \in [n, n + n_0)\}. $$

Common values of $n_0$ that will be considered are 1 and $+\infty$.

The following theorem states a general result concerning the local convergence of critical and sub-critical GW tree $\tau$ conditioned on $A_n$ toward Kesten's tree and its proof is in fact a straightforward adaptation of the proof of theorem 3.1 in [2].

We denote by

$$ \mathrm{dist}(\tau|\tau \in A_n) $$

the conditional law of $\tau$ given $\{\tau \in A_n\}$.

**Theorem 3.1.** Assume that assumption (1) hold, $\mu \le 1$ and that $\mathbb{P}(\tau \in A_n) > 0$ for $n$ large enough.

Then, if for all $\mathbf{t} \in \mathbb{T}_0$ and $x \in \mathcal{L}_0(\mathbf{t})$,

$$ (3) \qquad \liminf_{n \to +\infty} \frac{\mathbb{P}(\mathbf{t} \circledast_x \tau \in A_n)}{\mathbb{P}(\tau \in A_n)} \ge \frac{1}{\mu^{|x|}} $$

As $n \to +\infty$, We have

$$ \mathrm{dist}(\tau|\tau \in A_n) \to \mathrm{dist}(\tau^*). $$
*Proof.* Since $\mu \le 1$, we have a.s $\tau \in \mathbb{T}_0$ and $\tau^* \in \mathbb{T}_1$. So we will use Lemma 2.1 to prove the convergence.

Using [2] or else [4], we have for every $\mathbf{t} \in \mathbb{T}_0$, $x \in \mathcal{L}_0(\mathbf{t})$ and $\mathbf{t}' \in \mathbb{T}_0$:

$$ \mathbb{P}(\tau = \mathbf{t} \circledast_x \mathbf{t}') = \frac{\mathbb{P}(\tau = \mathbf{t})\mathbb{P}(\tau = \mathbf{t}')}{p(0)} $$

Let $\mathbf{t} \in \mathbb{T}_0$ and $x \in \mathcal{L}_0(\mathbf{t})$. For such $n$ we get:

$$ \begin{aligned} \mathbb{P}(\tau \in \mathbb{T}(\mathbf{t}, x) | \tau \in \mathbb{A}_n) &= \frac{1}{\mathbb{P}(\tau \in \mathbb{A}_n)} \sum_{\mathbf{t}' \in \mathbb{T}_0} \mathbb{P}(\tau = \mathbf{t} \circledast_x \mathbf{t}') 1_{\{\mathbf{t} \circledast_x \mathbf{t}' \in \mathbb{A}_n\}} \\ &= \frac{\mathbb{P}(\tau = \mathbf{t})}{p(0)\mathbb{P}(\tau \in \mathbb{A}_n)} \sum_{\mathbf{t}' \in \mathbb{T}_0} \mathbb{P}(\tau = \mathbf{t}') 1_{\{A(\mathbf{t} \circledast_x \mathbf{t}') \in \mathbb{A}_n\}} \end{aligned} $$

So, we obtain that

$$ \mathbb{P}(\tau \in \mathbb{T}(\mathbf{t}, x) | \tau \in \mathbb{A}_n) = \frac{\mathbb{P}(\tau = \mathbf{t}) \mathbb{P}(\mathbf{t} \circledast_x \tau \in \mathbb{A}_n)}{p(0) \mathbb{P}(\tau \in \mathbb{A}_n)} $$

Using (2), we deduce that:

$$ (4) \quad \mathbb{P}(\tau \in \mathbb{T}(\mathbf{t}, x) | \tau \in \mathbb{A}_n) = \mu^{|x|} \mathbb{P}(\tau^* \in \mathbb{T}(t, x)) \frac{\mathbb{P}(\mathbf{t} \circledast_x \tau \in \mathbb{A}_n)}{\mathbb{P}(\tau \in \mathbb{A}_n)} $$

Then, using (3), we deduce that:

$$ (5) \quad \liminf_{n \to +\infty} \mathbb{P}(\tau \in \mathbb{T}(\mathbf{t}, x) | \tau \in \mathbb{A}_n) \ge \mathbb{P}(\tau^* \in \mathbb{T}(t, x)) $$

Furthermore, for all $\mathbf{t} \in \mathbb{T}_0$ and all $n > A(\mathbf{t})$, we have

$$ \mathbb{P}(\tau = t, \tau \in \mathbb{A}_n) = \mathbb{P}(\tau = t, \mathbf{t} \in \mathbb{A}_n) \le 1_{\{\mathbf{t} \in \mathbb{A}_n\}} = 0 $$

and thus:

$$ \lim_{n \to +\infty} \mathbb{P}(\tau = t | \tau \in \mathbb{A}_n) = \mathbb{P}(\tau^* = t) = 0. $$

Finally, by Lemma 2.1, we have proved this result. □

<figure><img src="image_2.png" /></figure>

We have several remarks related to Theorem 3.1.

**Remark 3.2.** However, it seems that the above proof is almost copied from the proof of Theorem 3.1 in [2], it is easy to see that the sufficient condition in Theorem 3.1 is a particular case of this result.

**Remark 3.3.** We apply Theorem 3.1 to the Conditioning on the maximal out-degree: Let $\tau$ be a critical Galton-Watson tree and let $n$ be a large enough integer, we have for every $\mathbf{t} \in \mathbb{T}_0$, $x \in \mathcal{L}_0(\mathbf{t})$:

$$ \mathbb{P}(M(\mathbf{t} \circledast_x \tau) = n) = \mathbb{P}(M(\tau) = n) $$
This yields a short and an elementary proof of theorem 4.1 in [6]. In general, assume that $p$ is critical, $A$ satisfies the identity property in [3] and $\mathbb{P}(A(\tau) = n) > 0$ for any $n$, Then by theorem 3.1, as $n \to +\infty$,

$$ \mathrm{dist}(\tau|A(\tau) = n) \to \mathrm{dist}(\tau^*). $$

and as $n \to +\infty$,

$$ \mathrm{dist}(\tau|A(\tau) \ge n) \to \mathrm{dist}(\tau^*). $$

**Remark 3.4.** Assume that $p$ is critical. If $A$ satisfies the monotonicity property in [5], which is equivalent to $A(\mathbf{t} \circledast_x \mathbf{t}') \ge A(\mathbf{t}') \quad \forall \mathbf{t}' \in \mathbb{T}$, and $\mathbb{P}(A(\tau) \ge n) > 0$ for any integer $n$ (consider $n_0 = +\infty$), Then, using Theorem 3.1, we have as $n \to +\infty$:

$$ \mathrm{dist}(\tau|A(\tau) \ge n) \to \mathrm{dist}(\tau^*). $$

Thus, the monotonicity property is a particular case of our result, which provide another and a short proof of Theorem 2.1 in [5].

**Remark 3.5.** As a conclusion, Theorem 3.1 serves to cover all the different cases (identity, additivity and monotonicity) which allows to give a simple and short proof of theorem 2.2.1 and 2.2.4 in [3]

As direct application, we can recover several specific conditionings in the critical case:

(1) Conditioning on extinction after large time, see [7, 2], ($A = H$), as $n \to +\infty$

$$ \mathrm{dist}(\tau|H(\tau) \ge n) \to \mathrm{dist}(\tau^*). $$

(2) Conditioning on the total population size, ($A = \mathrm{Card}$), As $n \to +\infty$

$$ \mathrm{dist}(\tau|\mathrm{Card}(\tau) \ge n) \to \mathrm{dist}(\tau^*). $$

(3) Conditioning on the number of individuals having a given number of children:

Let $\mathcal{A}$ be a non-empty subset of $\mathbb{N}$. For a tree $\mathbf{t}$, we denote by $\mathbb{L}_{\mathcal{A}}(\mathbf{t})$ the total number of nodes in the tree $\mathbf{t}$ with outdegree in $\mathcal{A}$.

Assume that $\sum_{k \in \mathcal{A}} p(k) > 0$. Then as $n \to +\infty$

$$ \mathrm{dist}(\tau|\mathbb{L}_{\mathcal{A}}(\tau) \ge n) \to \mathrm{dist}(\tau^*). $$

(4) Conditioning on the number of protected nodes, see [1], Let $A(\mathbf{t})$ be the number of protected nodes in the tree $\mathbf{t}$. Then as $n \to +\infty$

$$ \mathrm{dist}(\tau|A(\tau) \ge n) \to \mathrm{dist}(\tau^*). $$
One of our original motivations for this result is the local convergence under the conditioning of large width. So right now we will apply Theorem 3.1 to this specific conditioning.

Since $p_0 + p_1 < 1$, we have $\mathbb{P}(L(\tau) \ge n) > 0$ for any $n$. Then Theorem 3.1 immediately gives the local convergence of critical GW trees to Kesten's tree, under the conditioning of large width.

Recall that the width $L(\mathbf{t})$ of a tree $\mathbf{t}$ is defined to be

$$L(\mathbf{t}) = \sup_{k \ge 0} \mathcal{Z}_k(\mathbf{t}).$$

**Corollary 3.6.** Assume that $p$ is critical. Then as $n \to +\infty$,

$$\mathrm{dist}(\tau|L(\tau) \ge n) \to \mathrm{dist}(\tau^*).$$

### 4. The conditioning on the largest generation, critical case

**Theorem 4.1.** Let $\tau$ be a critical GW tree with offspring distribution $p$ satisfying (1) with bounded support and let $\tau^*$ be a Kesten's tree associated to $p$. Let $\tau_n$ be a random tree distributed according to $\tau$ conditionally on $\{L(\tau) = n\}$. Then as $n \to +\infty$,

$$\mathrm{dist}(\tau_n) \to \mathrm{dist}(\tau^*).$$

where the limit is understood along the infinite sub-sequence $\{n \in \mathbb{N} : \mathbb{P}(L(\tau) = n) > 0\}$.

*Proof.* We consider $A(\mathbf{t}) = L(\mathbf{t})$ with $n_0 = 1$ that is

$$A_n = \{\mathbf{t} \in \mathbb{T}; L(\mathbf{t}) = n\}.$$

Since $p(0) + p(1) < 1$, the set $\{n \in \mathbb{N} : \mathbb{P}(L(\tau) = n) > 0\}$ is infinite.

Let $\mathbf{t} \in \mathbb{T}_0, x \in \mathcal{L}_0(\mathbf{t})$ and $n$ be a large enough integer. We consider

$$K = \sup\{n \in \mathbb{N}, p(n) > 0\} < +\infty$$

be the supremum of the support of $p$. We have

$$\forall |x| \le s \le H(\mathbf{t}) \quad \mathcal{Z}_s(\mathbf{t}) + \mathcal{Z}_{s-|x|}(\tau) \le L(\mathbf{t}) + K^{s-|x|} < n$$

We deduce that, $L$ satisfies the property identity, that is:

$$L(t *_x \tau) = n \iff L(\tau) = n$$

According to Theorem 3.1, we deduce the convergence and hence end the proof.

□
We can generalize these results concerning the width by considering:

Let $\mathcal{A}$ be a non-empty subset of $\mathbb{N}$. For a tree $\mathbf{t}$ and $s$ a non-negative integer, we write

$$ \mathcal{L}_{\mathcal{A}}^{(s)}(\mathbf{t}) = \{u \in \mathbf{t}; |u| = s \text{ and } k_u(\mathbf{t}) \in \mathcal{A}\} $$

the set of individuals, in generation $s$, whose number of children belongs to $\mathcal{A}$ and $\mathbb{L}_{\mathcal{A}}^{(s)}(\mathbf{t})$ its cardinal. We consider

$$ \mathcal{S}_{\mathcal{A}}(\mathbf{t}) = \sup_{s \ge 0} \mathbb{L}_{\mathcal{A}}^{(s)}(\mathbf{t}) $$

The case $\mathcal{A} = \{0\}$, $\mathbb{L}_{\mathcal{A}}^{(s)}(\mathbf{t})$ represents the number of leaves in generation $s$ of $\mathbf{t}$ and $\mathcal{S}_{\mathcal{A}}(\mathbf{t})$ the maximum number of leaves in the same generation. We can also have $\mathbb{L}_{\mathcal{A}}^{(s)}(\mathbf{t}) = \mathcal{Z}_s(\mathbf{t})$ and so $\mathcal{S}_{\mathcal{A}}(\mathbf{t}) = L(\mathbf{t})$ the largest generation by taking $\mathcal{A} = \mathbb{N}$.

Since $p(0) + p(1) < 1$ and $\sum_{k \in \mathcal{A}} p(k) > 0$, the set

$$ \{n \in \mathbb{N} : \mathbb{P}(\mathcal{S}_{\mathcal{A}}(\tau) = n) > 0\} \text{ is infinite.} $$

As, for all $\mathbf{t} \in \mathbb{T}_0$, $x \in \mathcal{L}_0(\mathbf{t})$ and $n$ be a large enough integer,

$$ \forall |x| \le s \le H(\mathbf{t}) \quad \mathbb{L}_{\mathcal{A}}^{(s)}(\mathbf{t}) + \mathbb{L}_{\mathcal{A}}^{(s-|x|)}(\tau) \le L(\mathbf{t}) + K^{s-|x|} < n $$

The same work as in the previous proof allows to prove

**Theorem 4.2.** Let $\tau$ be a critical GW tree with offspring distribution $p$ satisfying (1) with bounded support and such that $\sum_{k \in \mathcal{A}} p(k) > 0$. Let $\tau^*$ be a Kesten's tree associated to $p$. Then as $n \to +\infty$,

$$ \mathrm{dist}(\tau|\mathcal{S}_{\mathcal{A}}(\tau) = n) \to \mathrm{dist}(\tau^*). $$

where the limit is understood along the infinite sub-sequence $\{n \in \mathbb{N} : \mathbb{P}(\mathcal{S}_{\mathcal{A}}(\tau) = n) > 0\}$.

and

$$ \mathrm{dist}(\tau|\mathcal{S}_{\mathcal{A}}(\tau) \ge n) \to \mathrm{dist}(\tau^*). $$

Note that this type of conditioning is never treated.

**Acknowledgements.** Sincere thanks to anonymous referees for their comments and suggestions, which improved considerably the presentation of this paper.
REFERENCES

[1] R. Abraham, A. Bouaziz and J. F. Delmas, *Local limits of Galton-Watson trees conditioned by the number of the protected nodes*, Appl. J. of Probab., 54 (01).

[2] R. Abraham and J. F. Delmas, *Local limits of conditioned Galton-Watson trees: the infinite spine case*, Elec. J. of Probab., **19**: Article 2, 1-19, 2014.

[3] R. Abraham and J. F. Delmas, *An introduction to Galton-Watson trees and their local limits*, arxiv; 1506.05571,2015.

[4] Bertoin, J. (2013). *On largest offspring in a critical branching process with finite variance*, J. Appl. Prob. 50 791-800.

[5] X. He, *Local Convergence of Critical Random Trees and Continuous-State Branching Processes*, Journal of Theoretical Probability (2022) 35:685713.

[6] X. He, *Conditioning Galton-Watson trees on large maximal out-degree*, Journal of Theoretical Probability (2017) Springer, vol. 30(3), pages 842-851, September.

[7] H. Kesten, *Subdiffusive behavior of random walk on a random cluster*, Ann. Inst. H. Poincaré Probab. Statist. **22**(1986), no. 4, 425-487.

[8] J. Neveu, *Arbres et processus de Galton-Watson*, Ann. Inst. H. Poincaré Probab. Statist. **22** (1986), no. 2, 199-207.

AYMEN BOUAZIZ, INSTITUT PRÉPARATOIRE AUX ÉTUDES SCIENTIFIQUES ET TECHNIQUES, 2070 LA MARSA, TUNISIE.

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