Coding multitype forests: application to the law of the total population

# Coding multitype forests: application to the law of the total population of branching forests

Loïc Chaumont Loïc Chaumont – LAREMA – UMR CNRS 6093, Université d’Angers, 2 bd Lavoisier, 49045 Angers cedex 01  and  Rongli Liu Rongli Liu – Department of Mathematics, Nanjing University, Nanjing, 210093, P.R.China
July 14, 2019
###### Abstract.

By extending the breadth first search algorithm to any -type critical or subcritical irreducible branching forest, we show that such forests can be encoded through independent, integer valued, -dimensional random walks. An application of this coding together with a multivariate extension of the Ballot Theorem which is obtained here, allow us to give an explicit form of the law of the total population, jointly with the number of subtrees of each type, in terms of the offspring distribution of the branching process.

###### Key words and phrases:
Multitype branching forest, coding, random walks, ballot theorem, total population, cyclic exchangeablity.
###### 2010 Mathematics Subject Classification:
60C05, 05C05
This work was supported by MODEMAVE research project from the Région Pays de la Loire.
Ce travail a bénécié d’une aide de l’Agence Nationale de la Recherche portant la référence ANR-09-BLAN-0084-01.

## 1. Introduction

Let be the labeling in the breadth first search order of the vertices of a critical or subcritical branching forest with progeny distribution . Call , the size of the progeny of the -th vertex, then the stochastic process defined by,

 X0=0andXn+1−Xn=p(un+1)−1,n≥0

is a downward skip free random walk with step distribution , from which the entire structure of the original branching forest can be recovered. We will refer to this random walk as the Lukasiewicz-Harris coding path of the branching forest, see Section 6 of [8], Section 1.1 of [4] or Section 6.2 of [15]. A nice example of application of this coding is that the total population of the first trees of the forest, see Figure 1, may be expressed as the first passage time of at level , that is,

 Tk=inf{i:Xi=−k}.

This result combined with the following Kemperman’s identity (also known as the Ballot Theorem, see [16], Lemma 5 in [3] or Section 6.2 in [15]):

 P(Tk=n)=knP(Xn=−k),

allows us to compute the law of the total population of in terms of the progeny distribution . Note that the total population is actually a functional of the associated branching process, , since the random variable represents the number of individuals at the -th generation in the forest. The expression of this law was first obtained by Otter [14] and Dwass [5].

###### Theorem 1.1 (Otter (49) and Dwass (69)).

Let be a critical or subcritical branching process. Let be its law when it starts from and denote by its progeny law. Let be the total size of the population generated by , that is . Then for any ,

 (1.1) Pk(O=n)=knν∗n(n−k),

where is the -th iteration of the convolution product of the probability by itself.

More generally, whenever a functional of the branching forest admits a ’nice’ expression in terms of the Lukasiewicz-Harris coding path, we may expect to obtain an explicit form of its law. For instance, the law of the number of individuals with a given degree in the first trees can be obtained in this way. We refer to Proposition 1.6 in [9] where the law of the number of leaves, first obtained in [11], is derived from the Lukasiewicz-Harris coding.

The goal of this paper is to extend the above program to the multitype case. The Lukasiewicz-Harris coding will first be extended to multitype forests and will lead to the bijection stated in Theorem 2.7 between forests and some set of coding sequences. Then in order to obtain the multitype Otter-Dwass identity which is stated in Theorem 1.2, we first need the equivalent of the Ballot Theorem. This theorem together with its equivalent deterministic form, the multivariate Cyclic Lemma, are actually amongst the most important results of this paper. Both results require more preliminary notation and will be stated further in the text, see Lemma 3.3 and Theorem 3.4.

Let us first set some definitions and notation in multitype branching processes. We set and , and for any integer , the set will be denoted by . In all the sequel of this paper, will be an integer such that . On a probability space , we define a -type branching process , as a valued Markov chain with transition probabilities:

 P(Zn+1=(k1,…,kd)|Zn=(r1,…,rd))=ν∗r11∗⋯∗ν∗rdd(k1,…,kd),

where are distributions on and is the -th iteration of the convolution product of by itself, with . For , we will denote by the probability law . The vector will be called the progeny distribution of . According to this process, each individual of type gives birth to a random number of children with law , independently of the other individuals of its generation. The integer valued random variable is the total number of individuals of type , at generation . For , let us define the rate

 mij=∑z∈Zd+zjνi(z),

that corresponds to the mean number of children of type , given by an individual of type and let

 M:=(mij)i,j∈[d]

be the mean matrix of . Suppose that the extinction time is a.s. finite, that is

 (1.2) T=inf{n:Zn=0}<∞,a.s.

Then let be the total number of individuals of type which are born up to time (including individuals of the first generation):

 Oi=T∑n=0Z(i)n=∑n≥0Z(i)n.

The vector will be called the total population of the multitype branching process.

Up to now, most of the results on the exact law of the total population of multitype branching processes concern non irreducible, 2-type branching processes. Let us now recall them. In the case where and when and but , it may be derived from Theorem 1  in [3], that the distribution of the total population of is given by

 (1.3) P(r1,0)(O1=n1,O2=n2)=r1n1ν∗n11(n1−r1,n2),1≤r1≤n1.

When and but , after some elementary computation, combining the identities in (1.1) and (1.3), we obtain that for ,

 (1.4) P(r1,0)(O1=n1,O2=n2)=r1n1n2n2∑j=0jν∗n11(n1−r1,j)ν∗n22(0,n2−j).

Note that (1.3) and (1.4) concern only the reducible case, when and , a.s. As far as we know, those are the only situations where the law of the total population of multitype branching processes is known explicitly.

Recall that if is irreducible, then according to Perron-Frobenius Theorem, it admits a unique eigenvalue which is simple, positive and with maximal modulus. In this case, we will also say that is irreducible. If moreover, is non-degenarate, that is, if individuals have exactly one offspring with probability different from 1, then extinction, that is (1.2), holds if and only if , see [7], [12] and Chapter V of [1]. If , we say that is critical and if , we say that is subcritical. The results of this paper will be concerned by the case where is irreducible, non-degenarate, and critical or subcritical so that (1.2) holds, that is the multitype branching process becomes extinct with probability 1. However, let us emphasize that this assumption is only made for simplicity reasons and that all the proofs can be adapted to the case where the process is supercritical and/or reducible.

The next result gives the joint law of the total population together with the total number of individuals of type , whose parent is of type , , up to time . Let us denote by this random variable. We emphasize that the variables are not functionals of the multitype branching process . So, their formal definition and the computation of their law require a more complete information provided by the forest. Theorem 1.1 and identity (1.4) are extended as follows:

###### Theorem 1.2.

Assume that the -type branching process is irreducible, non-degenarate and critical or subcritical. For , let be the total number of individuals of type , up to the extinction time and for , let be the total number of individuals of type , whose parent is of type , up to time .

Then for all integers , , , , such that , , , for , , and ,

 Pr(O1=n1,…,Od=nd,Aij=kij,i,j∈[d],i≠j) = \rm det(K)¯n1¯n2…¯ndd∏i=1ν∗nii(ki1,…,ki(i−1),ni+kii,ki(i+1),…,kid),

where , , and is the matrix to which we removed the line and the column , for all such that .

Our proof of Theorem 1.2 uses a bijection, displayed in Theorem 2.7, between multitype forests and a particular set of multidimensional, integer valued sequences. A consequence of this result is that any critical or subcritical irreducible multitype branching forest is encoded by independent, -dimensional random walks, see Theorem 3.1. Then, in a similar way to the single type case, the total population, jointly with the number of subtrees of each type in the forest, is expressed as the first passage time of this multivariate process in some domain. The extension of the Ballot Theorem obtained in Theorem 3.4 allows us to conclude as in the single type case. Another analogy with the single type case is that the multivariate Lagrange inversion formula known as the Lagrange-Good formula, see [6], can be derived from Theorem 1.2 by applying this theorem to the generating function of the random vector .

This paper is organized as follows. Section 2 is devoted to deterministic multitype forests. In Subsection 2.1, we present the space of these forests and in Subsection 2.2, we define the space of the coding sequences and we obtain the bijection between this space and the space of multitype forests. This result is stated in Theorem 2.7. Then in Section 3, we define the probability space of multitype branching forests, we display their multitype Lukasiewicz-Harris coding in Theorem 3.1 and we prove its application to the total population that is stated in Theorem 1.2. This result requires a multivariate extension of the Ballot Theorem, see Theorem 3.4, whose proof bears on the crucial combinatorial Lemma 3.3. The latter is proved in Section 4.

## 2. Multitype forests

### 2.1. The space of multitype forests

A plane forest, is a directed planar graph with no loops , with a finite or infinite set of vertices , such that the outer degree of each vertex is equal to 0 or 1 and whose connected components, which are called the trees, are finite. A forest consisting of a single connected component is also called a tree. In a tree , the only vertex with outer degree equal to 0 is called the root of . It will be denoted by . The roots of the connected components of a forest are called the roots of . For two vertices and of a forest , if is a directed edge of , then we say that is a child of , or that is the parent of . The set of plane forests will be denoted by . The elements of will simply be called forests.

We will sometimes have to label the forests, which will be done in the following way. We first give an order to the trees of the forest and denote them by (we will usually write if no confusion is possible). Then each tree is labeled according to the breadth first search algorithm: we read the tree from its root to its last generation by running along each generation from the left to the right. This definition should be obvious from the example of Figure 1. If a forest contains at least vertices, then the -th vertex of is denoted by . When no confusion is possible, we will simply denote the -th vertex by .

Recall that is an integer such that . To each forest , we associate an application such that in the labeling defined above, if have the same parent, then . For , the integer is called the type (or the color) of . The couple is called a -type forest. When no confusion is possible, we will simply write . The set of -type forests will be denoted by . We emphasize that although there is an underlying labeling for each forest, and are sets of unlabeled forests. A 2-type forest is represented on Figure 2 below.

A subtree of type of a -type forest is a maximal connected subgraph of whose all vertices are of type . Formally, is a subtree of type of , if it is a connected subgraph whose all vertices are of type and such that either has no parent or the type of its parent is different from . Moreover, if the parent of a vertex belongs to , then . Subtrees of type of are ranked according to the order of their roots in and are denoted by . The forest is called the subforest of type of . It may be considered as an element of . We denote by the elements of , ranked in the breadth first search order of . The subforests of type 1 and 2 of a 2-type forest are represented in Figure 3.

To any forest , we associate the reduced forest, denoted by , which is the forest of obtained by aggregating all the vertices of each subtree of with a given type, in a single vertex with the same type, and preserving an edge between each pair of connected subtrees. An example is given in Figure 4.

### 2.2. Coding multitype forests

For a forest and , when no confusion is possible, we denote by the number of children of type of . For each , let be the number of vertices in the subforest of . Then let us define the -dimensional chain , with length and whose values belong to the set , by and if ,

 (2.5) xi,jn+1−xi,jn=pj(u(i)n+1),if i≠j andxi,in+1−xi,in=pi(u(i)n+1)−1,0≤n≤ni−1,

where we recall that is the labeling of the subforest in its own breadth first search order. Note that the chains , for are nondecreasing whereas is a downward skip free chain, i.e. , for . The chain corresponds to the Lukasiewicz-Harris coding walk of the subforest , as defined in the introduction, see also Section 6.2 in [15] for a proper definition. In particular, if is finite, then . These properties of the chains lead us to the following definition.

###### Definition 2.1.

Let be the set of -valued sequences, , such that for all , is a -valued sequence defined on some interval of integers, , , which satisfies and if then

• for , the sequence is nondecreasing,

• for all , , .

A sequence will sometimes be denoted by and for more convenience, we will sometimes denote by . The vector , where will be called the length of .

Relation (2.5) defines an application from the set to the set . Let us denote by this application, that is

 (2.6) Ψ:Fd → Sd (f,cf) ↦ Ψ((f,cf))=x.

For , set and define the first passage time process of the chain as follows:

 (2.7) τ(i)k=min{n≥0:xi,in=−k},0≤k≤ki,

where , if . If is the image by of a forest , i.e. , then is the (finite or infinite) number of trees in the subforest and for , the time is the total number of vertices which are contained in the first trees of , i.e. . This fact is well known and easily follows from the Lukasiewicz-Harris coding of the single type forest , see the introduction and Lemma 6.3 in [15]. Then for , define the integer valued sequence

 (2.8) ¯xi,jk=xi,j(τ(i)k),0≤k≤ki.

If , then we may check that when , is the number of subtrees of type whose root is the child of a vertex in . Or equivalently, it is the number of vertices of type whose parent is a vertex of . For each , we set

 ¯x(i)=(¯xi,1,…,¯xi,d)and¯x=(¯x(1),¯x(2),…,¯x(d)).

Clearly for , the sequence is nondecreasing and , for all and . Therefore and recalling the definition of the reduced forest, , see the end of Section 2.1, we may check that:

 (2.9) Ψ((fr,cfr))=¯x.

For a forest with trees , we will denote by the sequence of types of the roots of , i.e.

 c(f,cf):=(cf(r(t1)),cf(r(t2)),…).

Note that and that . When no confusion is possible, will simply be denoted by and we will call it the root type sequence of the forest.

Then before we state the general result on the coding of multitype forests in Theorem 2.7, we first need to show that the sequences together with allow us to encode the reduced forest , i.e. this forest can be reconstructed from . This claim is stated in Lemma 2.5 below. In order to prove it, we first need to describe the set of sequences which encode reduced forests and to state the preliminary Lemma 2.2 regarding these sequences.

Recall that and let us define the following (non total) order in : for two elements and of we write if for all . Moreover we write if and if there is such that . For an element of with length , and , we say that the system of equations admits a solution if there exists , such that and

 (2.10) rj+d∑i=1xi,j(si)=0,j=1,2,…,d.

We will see in Lemma 2.5 and Theorem 2.7 that for any finite forest with roots of type , the length of is a solution of and this solution is the smallest one in a sense that is specified in the following lemma.

###### Lemma 2.2.

Let and . Assume that the system admits a solution, then

• there exists a unique solution of the system such that if is any solution of , then . Moreover we have , for all . A solution such as will be called the smallest solution of the system .

• Let be such that . Then the system admits a solution. Let us denote its smallest solution by . Then the system , where , , admits a solution, and its smallest solution is .

A proof of this lemma is given in Section 4. For , with , that is , we define,

 Crd={c∈[d]r:Card{j∈[r]: cj=i}=ri,i∈[d]}.

We emphasize that the root type sequence of a forest with trees amongst which exactly trees have a root of type is an element of . Now we define the subsets of forests and reduced forests whose root type sequence is in and that contain at least one vertex of each type.

###### Definition 2.3.

Let , such that .

• We denote by , the subset of of forests with trees, which contain at least one vertex of each type, and such that .

• We denote by the subset of , of reduced forests. More specifically, if and if for each , vertices of type in have no child of type .

Then we define the sets of coding sequences related to and .

###### Definition 2.4.

Let be such that .

• We denote by the subset of of sequences whose length belongs to and corresponds to the smallest solution of the system defined in .

• We denote by the subset of consisting in sequences, such that , for all and .

Then we first establish a bijection between the sets and . Recall the definition of in (2.6).

###### Lemma 2.5.

Let be such that , then the mapping

 Φ:¯Frd → ¯Srd×Crd (f,cf) ↦ (Ψ((f,cf)),c(f,cf))

is a bijection.

###### Proof.

Let and let be the total number of subtrees of type which are contained in . (Note that since is a reduced forest, its subtrees are actually single vertices.) By definition, , for each . The fact that, follows from Definition 2.3 . Then let us show that . Since is a reduced forest, then . Besides, from (2.8), has length and for , is the number of subtrees of type whose root is a child of a vertex of , i.e. of any subtree of type in . Hence for , is the total number of subtrees of type in , whose root is a child of a vertex of type , . Then in order to obtain the total number of subtrees of type , it remains to add to , the number of subtrees of type whose root is one of the roots of , where . The latter number is , so that . Since moreover from (2.8), , for all , we have proved that is a solution of the system . It remains to prove that it is the smallest solution.

Let us first assume that , so that consists in a single tree whose root has color . Then we can reconstruct, this tree from the sequences , by inverting the procedure defined in (2.5) and this reconstruction procedure gives a unique tree. Indeed, by definition of the application , each sequence , is associated to a unique ’marked subforest’, say , of type whose vertices kept the memory of their progeny. More specifically, for , the increment gives the progeny of the -th vertex of the subforest . This connection between marked subforests and sequences is illustrated on Figure 5.

Now let be the smallest solution of the system . Let and suppose that we have been able to perform the reconstruction procedure until , that is from the sequences , . Then since is not a solution of , we see from what has been proved just above that the tree that is obtained is ’not complete’. That is, at least one of its leaves (say of type ) is marked, so that this leaf should still get children whose types and numbers are given by the next jump , for , according to the reconstruction procedure. Thus, doing so, we necessarily end up with a tree from the sequences , , and this tree is complete, that is none of its leaves is marked. Then since the reconstruction procedure obtained by inverting (2.5), gives a unique tree, we necessarily have .

Then let . Assume with no loss of generality that the root of the first tree of has color 1. Let be the number of subtrees of type in . From Lemma 2.2, the system , where , admits a smallest solution. Moreover from the reconstruction procedure which is described above, this solution is . Suppose now with no loss of generality that the second tree, in has color 2. Let be the number of subtrees of type in . Then from the same arguments as for the reconstruction of the first tree, may be reconstructed from the system , where and , . Moreover admits as a smallest solution. Then from part of Lemma 2.2, is the smallest solution of the system . So we have proved the result for the forest consisting of the trees and . Then by iterating these arguments for each tree of , we obtain that .

Conversely, let , and let be the smallest solution of the system . Then let us show that there is a forest such that and . Assume, without loss of generality that . From Lemma 2.2 , there is a smallest solution, say , to the system , where . Then we may reconstruct a unique forest (consisting in a single tree) such that and by inverting the procedure that is described in (2.5). Assume for instance that and set , then from Lemma 2.2 , there is a smallest solution, say , to the system . Moreover, is the smallest solution of the system , where , . Then as before, we can reconstruct a unique tree such that and such that the forest satisfies and . Then iterating these arguments, we may reconstruct a unique forest such that and .

Let with length and recall from (2.8), the definition of the associated sequence , with length , such that .

###### Lemma 2.6.

Let , such that and , with length and set , . If is the smallest solution of the system i.e. , then is the smallest solution of the system . Conversely, if