Scaling limits for a family of unrooted trees
We introduce weights on the unrooted unlabelled plane trees as follows: let be a probability measure on the set of nonnegative integers whose mean is bounded by ; then the -weight of a plane tree is defined as , where the product is over the set of vertices of . We study the random plane tree with a fixed diameter sampled according to probabilities proportional to these -weights and we prove that, under the assumption that the sequence of laws , , belongs to the domain of attraction of an infinitely divisible law, the scaling limits of such random plane trees are random compact real trees called the unrooted Lévy trees, which have been introduced in .
AMS 2010 subject classifications: Primary 60J80, 60E07. Secondary 60E10, 60G52, 60G55.
Keywords: random tree, unlabelled unrooted plane tree, Lévy trees, height process, diameter.
Models of random rooted trees have been extensively studied (see for instance Aldous , Devroye , Le Gall , Drmota , Janson ) often because of their connections with branching processes: an eminent example being the model of Galton–Watson trees. The scaling limits of Galton–Watson trees are Lévy trees, introduced by Le Gall and Le Jan . Lévy trees extend Aldous’ notion of Brownian Continuum Random Tree [5, 7]; they describe the genealogy of continuous-state branching processes; they are closely related to fragmentation and coalescent processes: see Miermont [32, 33], Haas & Miermont , Goldschmidt & Haas , Abraham & Delmas [1, 3]. Their probabilistic and fractal properties are studied in Duquesne & Le Gall [13, 14].
However, there are situations where unrooted trees arise naturally. In this work, we focus on unrooted and unlabelled plane trees that are graph-trees embedded into the oriented plane and considered up to orientation-preserving homeomorphisms (see Section 2 for a precise definition).
We propose here a model of random plane trees defined as follows. Let be a probability measure on . The -weight of a plane tree is the quantity:
where denotes the degree of the vertex in . These -weights induce a probability measure on the sets of plane trees with a fixed diameter. More precisely, the graph distance between two vertices of a tree is the number of edges on the unique path of between the two vertices. Then the diameter of is the maximum distance between any pair of vertices of . For , let be a random plane tree with diameter such that the probability of the event is proportional to , for all plane tree with diameter (see Section 3 below for a more careful definition of ).
The above definition of the random plane tree is inspired by simply generated trees introduced by Meir & Moon (). Our study of is, on the other hand, motivated by a previous work , where distributional properties of the diameters of general Lévy trees are examined. In particular, from the study there, a notion of unrooted Lévy trees naturally arises. Intuitively, an unrooted Lévy tree with diameter is obtained from two independent Lévy trees conditioned to have height by connecting their roots. It is shown in  that the spinal decomposition of an unrooted Lévy tree along its longest geodesic exhibits a remarkably simple form. On the other hand, a classical (rooted) Lévy tree can be obtained from the unrooted one by picking a uniform point as the root. (See  or Section 2.4 for the precise statements.) Because of all these properties, the model of unrooted Lévy trees seems to us an interesting object to study.
The work  deals with the continuum trees. Here, we consider the discrete counterpart (namely, ) and the convergence of the discrete trees when the diameters tend to infinity. In the main result (Theorem 3, see also Remark 3 there), we show that unrooted Lévy trees appear in the scaling limits of . This result could be viewed as the analog for unrooted plane trees to the Duquesne–Le Gall’s Theorem , which establishes the convergence of rescaled Galton–Watson trees to Lévy trees.
As an essential ingredient of the main proof, we also prove a limit theorem for Galton-Watson trees conditioned to have a fixed large height (Proposition 2), which might be of independent interest. This result extends a previous one due to Le Gall  in the Brownian case, which has been proved by a different method. The idea here is to perform a simple transformation on Galton–Watson trees, which consists in extracting a Galton–Watson tree conditioned to have a height from a Galton–Watson tree conditioned to have a height .
There are many possible ways to condition a random tree to be large. Previous works (see for example Aldous , Marckert & Miermont , Haas & Miermont ) mainly focus on random trees with a fixed progeny. Here, we condition trees by their heights or by their diameters, which are more adapted to the limit objects considered here. We refer to the paper  of Le Gall for a discussion on various conditionings and their connections with the excursion measures.
Another feather of the current work is that, unlike most of the previous works on the limit theorems of random trees, we look at unrooted trees rather than rooted ones. Due to their internal symmetries, unrooted trees turn out to be more difficult to deal with. Here, we employ the centers of a tree: our proof below of Theorem 3 relies on a decomposition of at its centers. After handling a technical point on the central symmetries, we show that asymptotically, this decomposition results in two independent rooted trees with fixed heights.
2 Preliminaries and notation
Before stating the main results of the paper, we recall here some notation and the definitions of the various classes of discrete trees that appear in the proofs of the main results. We wish to emphasize on differences and connections between plane trees and ordered rooted trees. We will see that equivalent classes of edge-rooted plane trees correspond to ordered rooted trees. On the other hand, we need to take into account the symmetry of a plane tree when performing a rooting of the tree. We also give a brief introduction to real trees and Lévy trees, upon which the construction of the limit objects relies. For a more extensive account on discrete trees, we refer to Drmota’s book ; for the technical background on Lévy trees we refer to Duquesne & Le Gall [13, 14]; see also Evans  and the references there for more information on real trees.
Unless otherwise specified, all the random variables that we mention here are defined on a common probability space .
2.1 Diameter and centers of trees
A tree is a connected graph without cycle. We only consider finite trees. Two vertices are adjacent if there is an edge between them. In that case, we write ; we also use the notation to indicate the oriented edge pointing from to . The degree of a vertex in a tree is the number of its adjacent vertices: . The size of a tree , denoted by , is the number of its vertices. A path from to in the tree is a sequence of adjacent vertices . We denote by the unique self-avoiding path joining to . Then the graph distance between and , denoted by , is the number of edges of , which we also refer to as the length of the path.
For a tree , we denote respectively by and the sets of its vertices and of its edges. For all , we set
We then define respectively
We say that is the diameter of the tree . The following notion plays an important role in this work.
|A vertex is a center of the tree if .||(2)|
In  (1869) C. Jordan proved that a tree has either one or two centers. More precisely, for the tree , we have a dichotomy in the number of centers of depending on the parity of its diameter.
Bi-centered case: if is odd, then and there are two adjacent centers . Moreover in this case, for any path such that , and are the two midpoints of the path. Namely, they are the only vertices in at distance from either or .
Uni-centered case: if is even, then and there is a unique center . Moreover in this case, for any path such that , belongs to the path and is situated at equal distance from and .
As it turns out, in our study of unrooted trees, centers are convenient choices for roots.
2.2 Ordered rooted trees and Galton-Watson trees
Let us first recall Ulam’s coding for ordered rooted trees. To this end, write . Define so that an element of is a finite sequence: for some integer and , . We say that a finite subset of is an ordered rooted tree if it satisfies the following three properties.
If , then .
For all , there exists a nonnegative integer such that if , then , for all .
Alternatively, an ordered rooted tree can be identified as a family tree: is the common ancestor; the elements form the first generation, ranked in their birth orders; more generally, an individual has exactly children, namely , . Observe that the ancestors of are given by , .
We denote by the set of ordered rooted trees. Let us recall a well-known result on the enumeration of ordered rooted trees. For each , the number of ordered rooted trees of size is given by , the -th Catalan number.
Let be a probability distribution on . Since we only consider finite trees, let us assume that the mean of is bounded by . We also suppose that the support of is not contained in the set to exclude trivial cases. We summarize our assumptions on as follows:
For each finite ordered rooted tree , we set
Standard arguments (see for example Neveu ) show that defines a probability measure on ; is then called Galton–Watson law with offspring distribution . In what follows, a GW()-tree refers to a random variable, say , defined on the probability space taking values in such that , for all .
2.3 Plane trees
Plane trees are graph-trees embedded into the oriented plane , considered up to orientation preserving homeomorphisms from to . More precisely, an oriented edge in (distinct from a loop) is a continuous and injective function , considered up to reparametrization (i.e. strictly increasing homeomorphisms from to ). The tail of the oriented edge is and its head is . The reversed edge is , , also considered up to reparametrization. A non-oriented edge is then given by . The endpoints of are and its inner part is . Note that the endpoints and the inner part of an edge do not depend on any particular parametrization. An embedded tree in the plane is a pair such that
is a connected subset of formed by a finite set of edges (as defined above) whose inner parts are pairwise disjoint and do not intersect and whose endpoints belong to ;
Two embedded plane trees and are said to be equivalent if there exists an orientation preserving homeomorphism such that and . The equivalence class of an embedded plane tree is referred to as a plane tree in the rest of the paper. We denote by the set of plane trees.
A plane tree naturally induces a graph-tree; but it also carries additional structures inherited from the oriented plane. In particular, for each vertice , the orientation of the plane induces a cyclic order on the set of vertices that are adjacent to (namely, the set of the neighbors of ); see Figure 1. This notion of cyclic order will be useful in what follows.
In , Harary, Prins & Tutte deduce a functional equation for the generating function of the numbers of plane trees with given sizes, which eventually leads to the following closed formula due to Walkup : for each ,
Let us mention that plane trees are particular instances of planar maps: they are planar maps with one face. We refer to Mohar & Thomassen’s book  for a more precise account on embedded graphs on surfaces and to Lando & Zvonkin’s book  for a combinatorial definition of planar maps.
Edge-rooted plane trees and ordered rooted trees.
A plane tree with a distinguished oriented edge is called an edge-rooted plane tree. Two edge-rooted plane trees and are said to be equivalent if there exists an orientation preserving homeomorphism satisfying . Let be an edge-rooted plane tree. We can associate with it an ordered rooted tree in the following way (see also Fig. 1). Let us employ the terminology of family tree and recall that there is a cyclic order on the neighbors of each vertex of which is induced by the orientation of the plane. Let , the tail of . We view as the common ancestor. Let be the neighbors of ordered in such a way that and that is next to in the cyclic order, for all . Then the sequence forms the first generation ranked in their birth orders. More generally, for a vertex , let us write for its neighbors ordered in such a way that is the unique vertex of that is adjacent to and that is next to in the cyclic order, for all . Then are the children of , ranked in their birth orders, while is the parent of . In this way, we can readily associate with an ordered rooted tree which we denote by . It is straightforward to check that if and are two equivalent edge-rooted plane trees, then . On the other hand, for an ordered rooted tree satisfying , we can always find an embedded plane tree and an oriented edge of such that . To sum up, there is a bijection between the set of equivalence classes of edge-rooted plane trees and the set of ordered rooted trees with size .
At this point, let us make an important remark on the number of possible ways in rooting a plane tree, which turns out to be one of the technical points that we need to deal with in the proof of the main theorem.
Due to a potential internal symmetry of the tree, the mapping is surjective but not injective in general, since different choices of edges may give rise to equivalent edge-rooted plane trees; see Figure 2 for an example. Indeed, let us observe from (5) that for all , we have
On the other hand, note that
which suggests that a “typical” large plane tree has no internal symmetries.
Contour functions of edge-rooted plane trees and ordered rooted trees.
We will use contour functions to study trees, whose definition we recall here. In the first place, let be an ordered rooted tree embedded into the plane in such a way that the common ancestor is located at the origin and the children of each vertex appear from left to right in increasing order of their birth orders. In a less formal way, imagine that each edge is a line segment of length and a particle explores the embedded tree with speed from the common ancestor, in a left-to-right way, backtracking as least as possible. It terminates its exploration at time , during which each edge is visited exactly twice by the particle. Denote by the distance of the particle from at time . Then the continuous function is called the contour function of the ordered rooted tree . See Fig. 3. We refer to Duquesne  for a formal definition. Note that is characterized by its contour function. In particular, the graph distance of can be found as follows. For , denote by the vertex of visited by the particle at time . Then, for all integers such that , we have
Next, we recall from above that an edge-rooted plane tree can be associated with an ordered rooted tree . Then we define the contour function of the edge-rooted tree to be that of and we denote it by .
2.4 Lévy trees
The class of Lévy trees is introduced by Le Gall & Le Jan  (see also Duquesne & Le Gall ). It extends Aldous’ Continuum Random Tree [5, 6, 7], which will sometimes be referred to as the Brownian case. Lévy trees are random compact metric spaces, or more specifically, random compact real trees. Let us first recall the definition of real trees.
One way to generalize graph-trees is to consider geodesic metric spaces without loops. More precisely, a metric space is a real tree if the following two properties hold.
For all , there is a unique isometric mapping such that and . In this case, let us write , the geodesic from to .
For all injective continuous functions , we have .
Alternatively, real trees are characterized by the four-point inequality: a connected metric space is a real tree if and only if for all ,
See Evans  for more details. In this work, we only consider compact real trees.
A rooted real tree is a real tree with a distinguished point , called the root. The degree of a point , denoted by , is the (possibly infinite) number of connected components of . Then, is said to be a leaf if , a simple point if and a branch point if .
Centers of compact real trees.
Let be a compact real tree. We set
The above definitions make sense as is compact. If is rooted at , then corresponds to the total height of the rooted tree . In that case, we simply denote . Note that is the diameter of the metric space . Recall from (2) Jordan’s definition for the centers of discrete trees. We introduce an analogous definition for real trees:
We have the following fact about the centers of compact real trees, which is the analog of Jordan’s theorem mentioned above.
Let be a compact real tree. The following statements hold true:
we have ;
there exists a unique center of ;
for all pairs of points such that , we have and .
Proof: since is compact, there exist such that . Let be such that . Let be an arbitrary point. We apply the four-point inequality (6) with , , and and we get after some simplification that
This entails that for all . It follows that . On the other hand, since , (9) entails that for any . It follows that . Combined with the previous argument, we obtain that . Thus, is a center of . Next, observe that if is a center of , then . Then we get from (9) that . This then entails the statements in b) and c) and thus completes the proof of the lemma.
The coding of real trees.
Let us briefly recall how real trees can be obtained from continuous nonnegative functions. To that end, we write for the space of -valued continuous functions equipped with the Polish topology of the uniform convergence on every compact subset of . Let us denote by the canonical process on . We are concerned with the case where has a compact support, and is distinct from the null function. We call such a function a coding function. Let us assume that is a coding function. Set , the lifetime of the coding function . Note that under our assumptions. For every , we set
It is straightforward to check that satisfies the four-point inequality (6). Note that is a pseudo-metric. We then introduce the equivalence relation by setting iff . Let
Standard arguments show that induces a metric on the quotient set , which we keep denoting by . Let be the canonical projection. Since is continuous, so is and is therefore a compact connected metric space which further satisfies the four-point inequality: it is a compact real tree. We also set to be the root of . See Duquesne  for more details on the coding of real trees by functions.
In what follows, we sometimes need to perform a re-rooting operation on the rooted real trees. In terms of the coding functions, this operation corresponds to the following path transformation, which we recall from Duquesne & Le Gall . Let be a coding function as defined above and recall that its lifetime . For any , denote by the unique element of such that is an integer multiple of . If , we define a coding function as follows:
It is not difficult to see that we can identify as the re-rooted tree .
Height processes of Lévy trees and excursion measures.
In , Le Gall & Le Jan introduce the height processes, which are the coding functions of Lévy trees (see also Duquesne & Le Gall ). We recall here the definition of height processes from their works. We refer the reader to Bertoin’s book  for background on Lévy processes. Let be a spectrally positive Lévy process starting from defined on the probability space . Then its law is characterized by its Laplace exponent in the sense that , for all . We assume that does not drift to . In this case, has finite expectation and takes the following Lévy-Khintchine form:
where and is a sigma-finite measure on satisfying . We restrict our attention to the case where satisfies the following assumption:
Under this assumption, there exists a continuous nonnegative process such that for all , the following limit holds in -probability:
where . The process is called the -height process. In the special case where (i.e. the Brownian case), classical arguments show that is distributed as a reflected Brownian motion.
It turns out that encodes a sequence of random real trees: each excursion of above zero corresponds to a tree in this sequence. More precisely, for any , we set . Basic results of the fluctuation theory entail that is a -valued strong Markov process, that is regular for and recurrent. Moreover, is a local time at for . We denote by the corresponding excursion measure of above . We can derive from (14) that only depends on the excursion of above which straddles . Moreover, we have , that is, the excursion intervals of above coincide with those of above . Let us denote by , , these excursion intervals. Set , . Then the point measure is a Poisson point measure on with intensity . Here, we have slightly abused the notation by letting stand for the “distribution” of under , which is a sigma-finite measure on . In the Brownian case, up to a multiplicative constant, is the Ito’s positive excursion measure of Brownian motion and reduces to the Poisson decomposition of a reflected Brownian motion above .
In the rest of the paper, we will work exclusively with the -height process under the excursion measure . The following holds true:
This shows that under is a coding function as defined above. Duquesne & Le Gall  then define the -Lévy tree as the real tree coded by under in the sense of (10). In that case, when there is no risk of confusion, we simply write instead of .
Lévy trees conditioned by their total heights.
Let be the -height process under its excursion measure , as defined above. We use the notation , which coincides with the total height of the -Lévy tree . Let us recall from Duquesne & Le Gall  (Corollary 1.4.2) the following distributional properties of .
also -a.e., there exists a unique such that .
Abraham & Delmas in  define the laws of the -height process conditioned by the total heights. More precisely, they construct a family of probability laws , , on which satisfy the following properties:
The mapping is continuous with respect to the weak topology on .
In addition, the authors of  give a Poisson decomposition at the unique maximum point of , which generalizes Williams’ decomposition for Brownian excursions.
Lévy trees conditioned by their diameters.
Recall that stands for the Lévy tree coded by the -height process under its excursion measure . In , Duquesne & W. study the diameter of as well as a spinal decomposition of along its longest geodesic. In particular, the following results can be found there. We have that -a.e. there exists a unique pair of times such that . The distribution of under is given by
where is defined in (15). Next, we introduce the laws of conditioned by its diameter. To that end, let be two coding functions. The concatenation of and is the coding function defined as
For all , we define as the law of under . Namely, for all measurable functions ,
Then has the following properties.
-a.s. we have and there exists a unique pair of points such that . Moreover, the unique center of has degree : it is a simple point.
For all , . Moreover, the mapping is weakly continuous and for all measurable functions and ,
where the rerooting of at , defined in (11).
It follows from (18) that we have a regular version of the conditional laws that are obtained from by a uniform re-rooting: for all measurable functions , we have
We call the law of unrooted -Lévy trees with diameter . For a more extensive account, we refer to Duquesne & W. , Theorem 1.2 and Remark 1.2.
3 Main results
Recall that all the random variables here are defined on the same probability space . In particular, it contains the following.
For all positive integers , let be a probability law on that verifies (3). Let be a sequence of independent random variables with common law . We assume that there exists a non decreasing sequence of positive integers such that the following convergence holds in distribution for -valued random variables:
We set , , the generating function of . Let us denote by the -th iteration of , that is, and , . We assume that for all ,
Let be a GW()-tree. Recall from Section 2.3 the contour function of the ordered rooted tree . For convenience, we extend the definition of to by setting for all . We also set
which coincides with the total height of . Recall from Section 2.4 the -height process defined under the excursion measure . Under the assumptions (20) and (21), Duquesne & Le Gall in  have shown a general invariance principle for ; see Theorem 2.3.1 and Corollary 2.5.1 there. Then, they deduce (Proposition 2.5.2 of ) that for all ,
where the convergence holds in distribution on . Here, we show that their result can be extended to the following.
Let be given in (12) and satisfy (13). Let be a probability law on which verifies (3), for each . Suppose that (20) and (21) take place. Let be as in (20) and let . Let be the extended contour function of the GW-tree as defined above. Then, the following convergence holds in distribution on :
Here, stands for the law of the -height process conditioned to have a total height and stands for its lifetime.
The assumptions (20) and (21) are minimal for (22) to hold: see the discussion right after Theorem 2.3.1 in , p. 55. Let us also mention that if for all and (20) hold, then (21) is automatically verified. In this case, the limit in (20) is necessarily distributed as a spectrally positive -stable random variable, for some . See Theorem 2.3.2 in  for the details.
The aim of this work is to prove a limit theorem for a family of random unrooted unlabelled plane trees which are defined as follows. Let be a probability law on which satisfies (3). For a plane tree , we set
Recall that stands for the diameter of a tree . For all , let us denote
The above Proposition 2 plays an important role in our study of . Indeed, by rooting plane trees at their central edges (see the definition below), we will see (in Lemma 9) that is closely related to the laws of Galton–Watson trees conditioned by total heights.
Let be a plane tree with vertex set . Recall from (2) Jordan’s definition for the centers of a tree.
If is odd, then has two adjacent centers . In this case, we define the set of central edges of as