# The Probabilities of Trees and Cladograms under Ford’sα-Model

Authors: Tomás M. Coronado; Arnau Mir; Francesc Rosselló
Journal: The Scientific World Journal
 (2018)

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## Abstract

Ford’sα-model is one of the most popular random parametric models of bifurcating phylogenetic tree growth, having as specific instances both the uniform and the Yule models. Its general properties have been used to study the behavior of phylogenetic tree shape indices under the probability distribution it defines. But the explicit formulas provided by Ford for the probabilities of unlabeled trees and phylogenetic trees fail in some cases. In this paper we give correct explicit formulas for these probabilities.


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## Body

## 1. Introduction

The study of random growth models of rooted phylogenetic trees and the statistical properties of the shapes of the phylogenetic trees they produce was initiated almost one century ago by Yule [1] and it has gained momentum in the last 20 years: see, for instance, [2–8]. The final goal of this line of research is to understand the relationship between the forces that drive evolution and the topological properties of “real-life” phylogenetic trees [3, 9]; see also [10, Chapter 33]. One of the most popular such models is Ford’s α-model for rooted bifurcating phylogenetic trees orcladograms [4]. It consists of a parametric model that generalizes both the uniform model (where new leaves are added equiprobably to any arc, giving rise to the uniform probability distribution on the sets of cladograms with a fixed set of taxa) and Yule’s model (where new leaves are added equiprobably only topendant arcs, i.e., to arcs ending in leaves) by allocating a possibly different probability (that depends on a parameter α and hence its name, “α-model”) to the addition of the new leaves to pendant arcs or to internal arcs.When models like Ford’s model are used to contrast topological properties of phylogenetic trees contained in databases like TreeBase (https://treebase.org), only their general properties (moments, asymptotic behavior) are employed. But, in the course of a research where we have needed to compute the probabilities of several specific cladograms under this model [11], we have noticed that the explicit formulas that Ford gives in [4, §3.5] for the probabilities of cladograms and oftree shapes (unlabeled rooted bifurcating trees) are wrong, failing for some trees with n⩾8 leaves; see Propositions 29 and 32 in [4], with the definition of q^ given in page 30 therein, for Ford’s formulas.So, to help the future user of Ford’s model, in this paper we give the correct explicit formulas for these probabilities. This paper is accompanied by the GitHub pagehttps://github.com/biocom-uib/prob-alpha where the interested reader can find a SageMath [12] module to compute these probabilities and their explicit values on the sets Tn of cladograms with n leaves labeled 1,…,n, for every n from 2 to 8.


## 2. Preliminaries

### 2.1. Definitions, Notations, and Conventions

Throughout this paper, by atreeT, we mean a rooted bifurcating tree. As it is customary, we understand T as a directed graph, with its arcs pointing away from the root, which we shall denote by rT. Then, all nodes in T have out-degree either 0 (itsleaves, which form the set L(T)) or 2 (itsinternal nodes, which form the set Vint(T)). Thechildren of an internal node v are those nodes u such that (v,u) is an arc in T, and they form the set child(v). A node x is adescendant of a node v when there exists a directed path from v to x in T. For every node v, thesubtreeTvofTrooted atv is the subgraph of T induced on the set of descendants of v.A treeT isordered when it is endowed with anordering≺v on every set child(v). Acladogram (resp., anordered cladogram) on a set of taxa Σ is a tree (resp., an ordered tree) with its leaves bijectively labeled in Σ. Whenever we want to stress the fact that a tree is not a cladogram, that is, it is an unlabeled tree, we shall use the termtree shape.It is important to point out that although ordered trees have no practical interest from the phylogenetic point of view, because the orderings on the sets of children of internal nodes do not carry any phylogenetic information, they are useful from the mathematical point of view, because the existence of the orderings allows one to easily prove certain extra properties that can later be translated to the unordered setting (cf. Proposition1).Anisomorphism of ordered trees is an isomorphism of rooted trees that moreover preserves these orderings. Anisomorphism of cladograms (resp., of ordered cladograms) is an isomorphism of trees (resp., of ordered trees) that preserves the leaves’ labels. We shall always identify a tree shape, an ordered tree shape, a cladogram, or an ordered cladogram, with its isomorphism class, and in particular we shall make henceforth the abuse of language of saying that two of these objects, T,T′,are the same, in symbols T=T′, when they are (only) isomorphic. We shall denote by Tn∗ and OTn∗, respectively, the sets of tree shapes and of ordered tree shapes with n leaves. Given any finite set of taxa Σ, we shall denote by TΣ and OTΣ, respectively, the sets of cladograms and of ordered cladograms on Σ. When the specific set Σ is unrelevant and only its cardinal matters, we shall write Tn and OTn (with n=Σ) instead of TΣ and OTΣ, and then we shall understand that Σ is [n]=1,2,…,n.There exist natural isomorphism-preserving forgetful mappingsthat “forget” the orderings or the labels of the trees. In particular, we shall call the image of a cladogram underπ itsshape. Figure 1 depicts an example of images under these forgetful mappings.Figure 1
An example of images under the forgetful mappings between (ordered and unordered) cladograms and tree shapes. In the ordered objects, the ordering is represented by the nodes’ colors: gray≺ white.Let us introduce some more notations. For every nodev in a tree T, κT(v) is its number of descendant leaves. For every internal node v in an ordered tree T, with children v1≺vv2, itsnumerical split is the ordered pair NST(v)=(κT(v1),κT(v2)). If, instead, T is unordered and if child(v)=v1,v2 with κT(v1)⩽κT(v2), then NST(v)=(κT(v1),κT(v2)). In both cases, themultiset of numerical splits of T is NS(T)={NST(v)∣v∈Vint(T)}. For instance, if T is the cladogram depicted in Figure 2, then (1)NST=1,1,1,1,1,1,2,2,1,4,2,5.Figure 2
A cladogram inT7. The black nodes are its symmetric branch points.Asymmetric branch point in a tree T is an internal node v such that if v1 and v2 are its children, then the subtrees Tv1 and Tv2 of T rooted at them have the same shape. For instance, the symmetric branch points in the cladogram depicted in Figure 2 are those filled in black.Given two cladogramsT and T′ on Σ and Σ′, respectively, with Σ∩Σ′=∅, theirroot join is the cladogram T⋆T′ on Σ∪Σ′ obtained by connecting the roots of T and T′ to a (new) common root r; see Figure 3. If T,T′ are ordered cladograms, T⋆T′ is ordered by inheriting the orderings on T and T′ and ordering the children of the new root r as rT≺rrT′. If T and T′ are tree shapes, a similar construction yields a tree shape T⋆T′; if they are moreover ordered, then T⋆T′ becomes an ordered tree shape as explained above.Figure 3
The root joinT⋆T′.


### 2.2. Theα-Model

Ford’sα-model [4] defines, for every n⩾1, a family of probability density functions Pα,n(∗) on Tn∗ that depends on one parameter α∈[0,1], and then it translates this family into three other families of probability density functions Pα,n on Tn, Pα,n(o,∗) on OTn∗, and Pα,n(o) on OTn, by imposing that the probability of a tree shape is equally distributed among its preimages under π, πo,∗, and π∘πo=πo,∗∘π∗, respectively.It is well known [13] that every T∈Tn can be obtained in a unique way by adding recurrently to a single node labeled 1 new leaves labeled 2,…,n to arcs (i.e., splitting an arc (u,v) into two arcs (u,w) and (w,v) and then adding a new arc from the inserted node w to a new leaf) or to a new root (i.e., adding a new root w and new arcs from w to the old root and to a new leaf). The value of Pα,n(∗)(T∗) for T∗∈Tn∗ is determined through all possible ways of constructing cladograms with shape T∗ in this way. More specifically,(1)
ifT1 and T2 denote, respectively, the only cladograms in T1 and T2, let Pα,1′(T1)=Pα,2′(T2)=1;(2)
for everym=3,…,n, let Tm∈Tm be obtained by adding a new leaf labeled m to Tm-1. Then (2)Pα,m′Tm=αm-1-α·Pα,m-1′Tm-1if  the  new  leaf  is  added  to  an  internal  arc  or  to  a  new  root1-αm-1-α·Pα,m-1′Tm-1if  the  new  leaf  is  added  to  a  pendant  arc;(3)
When the desired numbern of leaves is reached, the probability of every tree shape Tn∗∈Tn∗ is defined as (3)Pα,n∗Tn∗=∑πTn=Tn∗Pα,n′Tn.For instance, Figure4 shows the construction of two cladograms in T5 with the same shape and how their probability Pα,5′ is built using the recursion in Step (2). If we generate all cladograms in T5 with this shape, we compute their probabilities Pα,5′, and then we add up all these probabilities, we obtain the probability Pα,5(∗) of this shape, which turns out to be 2(1-α)/(4-α); cf. [4, Figure 23].Figure 4
Two examples of computations of the probabilityPα,n′ of a cladogram through its construction in Step (2) of the definition of the α-model.OncePα,n(∗) is defined on Tn∗, it is transported to Tn, OTn∗, and OTn by defining the probability of an object in one of these sets as the probability of its image in Tn∗ divided by the number of preimages of this image:(i)
For everyT∈Tn, if π(T)=T∗∈Tn∗ and it has k symmetric branch points, then(4)Pα,nT=2kn!·Pα,n∗T∗,becauseπ-1T∗=n!/2k (see, e.g., [4, Lemma 31]).(ii)
For everyTo∈OTn, if πo(To)=T∈Tn, then(5)Pα,noTo=12n-1·Pα,nT,becauseπo-1T=2n-1 (T has 2n-1 different preimages under πo, obtained by taking all possible different combinations of orderings on the n-1 sets child(v), v∈Vint(T∗)).(iii)
For everyTo∗∈OTn∗, if πo,∗(To∗)=T∗∈Tn∗ and it has k symmetric branch points, then(6)Pα,no,∗To∗=12n-k-1·Pα,n∗T∗,becauseπo,∗-1T∗=2n-1-k (from the 2n-1 possible preimages of T∗ under πo,∗, defined by all possible different combinations of orderings on the n-1 sets child(v), v∈Vint(T∗), those differing only on the orderings on the children of the k symmetric branch points are actually the same ordered tree shape).The family(Pα,n(o,∗))n of probabilities of ordered tree shapes satisfies the useful Markov branching recurrence (in the sense of [2, §4]) given by the following proposition. In it and in the sequel, let, for every a,b∈Z+, (7)qαa,b=ΓαaΓαbΓαa+b·φαa,b,where (8)φαa,b=α2a+ba+1-2αa+b-2a-1and Γα:Z+→R is the mapping defined by Γα(1)=1 and, for every n⩾2, Γα(n)=(n-1-α)·Γα(n-1).Proposition 1.
For every0<m<n and for every Tm∗∈OTm∗ and Tn-m∗∈OTn-m∗, (9)Pα,no,∗Tm∗⋆Tn-m∗=qαm,n-mPα,mo,∗Tm∗Pα,n-mo,∗Tn-m∗.This recurrence, together with the fact thatPα,1(o,∗) of a single node is 1, implies that, for every To∗∈OTn∗,(10)Pα,no,∗To∗=∏a,b∈NSTo∗qαa,b.For proofs of Proposition 1 and (10), see Lemma 27 and Proposition 28 in [4], respectively.


## 2.1. Definitions, Notations, and Conventions

Throughout this paper, by atreeT, we mean a rooted bifurcating tree. As it is customary, we understand T as a directed graph, with its arcs pointing away from the root, which we shall denote by rT. Then, all nodes in T have out-degree either 0 (itsleaves, which form the set L(T)) or 2 (itsinternal nodes, which form the set Vint(T)). Thechildren of an internal node v are those nodes u such that (v,u) is an arc in T, and they form the set child(v). A node x is adescendant of a node v when there exists a directed path from v to x in T. For every node v, thesubtreeTvofTrooted atv is the subgraph of T induced on the set of descendants of v.A treeT isordered when it is endowed with anordering≺v on every set child(v). Acladogram (resp., anordered cladogram) on a set of taxa Σ is a tree (resp., an ordered tree) with its leaves bijectively labeled in Σ. Whenever we want to stress the fact that a tree is not a cladogram, that is, it is an unlabeled tree, we shall use the termtree shape.It is important to point out that although ordered trees have no practical interest from the phylogenetic point of view, because the orderings on the sets of children of internal nodes do not carry any phylogenetic information, they are useful from the mathematical point of view, because the existence of the orderings allows one to easily prove certain extra properties that can later be translated to the unordered setting (cf. Proposition1).Anisomorphism of ordered trees is an isomorphism of rooted trees that moreover preserves these orderings. Anisomorphism of cladograms (resp., of ordered cladograms) is an isomorphism of trees (resp., of ordered trees) that preserves the leaves’ labels. We shall always identify a tree shape, an ordered tree shape, a cladogram, or an ordered cladogram, with its isomorphism class, and in particular we shall make henceforth the abuse of language of saying that two of these objects, T,T′,are the same, in symbols T=T′, when they are (only) isomorphic. We shall denote by Tn∗ and OTn∗, respectively, the sets of tree shapes and of ordered tree shapes with n leaves. Given any finite set of taxa Σ, we shall denote by TΣ and OTΣ, respectively, the sets of cladograms and of ordered cladograms on Σ. When the specific set Σ is unrelevant and only its cardinal matters, we shall write Tn and OTn (with n=Σ) instead of TΣ and OTΣ, and then we shall understand that Σ is [n]=1,2,…,n.There exist natural isomorphism-preserving forgetful mappingsthat “forget” the orderings or the labels of the trees. In particular, we shall call the image of a cladogram underπ itsshape. Figure 1 depicts an example of images under these forgetful mappings.Figure 1
An example of images under the forgetful mappings between (ordered and unordered) cladograms and tree shapes. In the ordered objects, the ordering is represented by the nodes’ colors: gray≺ white.Let us introduce some more notations. For every nodev in a tree T, κT(v) is its number of descendant leaves. For every internal node v in an ordered tree T, with children v1≺vv2, itsnumerical split is the ordered pair NST(v)=(κT(v1),κT(v2)). If, instead, T is unordered and if child(v)=v1,v2 with κT(v1)⩽κT(v2), then NST(v)=(κT(v1),κT(v2)). In both cases, themultiset of numerical splits of T is NS(T)={NST(v)∣v∈Vint(T)}. For instance, if T is the cladogram depicted in Figure 2, then (1)NST=1,1,1,1,1,1,2,2,1,4,2,5.Figure 2
A cladogram inT7. The black nodes are its symmetric branch points.Asymmetric branch point in a tree T is an internal node v such that if v1 and v2 are its children, then the subtrees Tv1 and Tv2 of T rooted at them have the same shape. For instance, the symmetric branch points in the cladogram depicted in Figure 2 are those filled in black.Given two cladogramsT and T′ on Σ and Σ′, respectively, with Σ∩Σ′=∅, theirroot join is the cladogram T⋆T′ on Σ∪Σ′ obtained by connecting the roots of T and T′ to a (new) common root r; see Figure 3. If T,T′ are ordered cladograms, T⋆T′ is ordered by inheriting the orderings on T and T′ and ordering the children of the new root r as rT≺rrT′. If T and T′ are tree shapes, a similar construction yields a tree shape T⋆T′; if they are moreover ordered, then T⋆T′ becomes an ordered tree shape as explained above.Figure 3
The root joinT⋆T′.


## 2.2. Theα-Model

Ford’sα-model [4] defines, for every n⩾1, a family of probability density functions Pα,n(∗) on Tn∗ that depends on one parameter α∈[0,1], and then it translates this family into three other families of probability density functions Pα,n on Tn, Pα,n(o,∗) on OTn∗, and Pα,n(o) on OTn, by imposing that the probability of a tree shape is equally distributed among its preimages under π, πo,∗, and π∘πo=πo,∗∘π∗, respectively.It is well known [13] that every T∈Tn can be obtained in a unique way by adding recurrently to a single node labeled 1 new leaves labeled 2,…,n to arcs (i.e., splitting an arc (u,v) into two arcs (u,w) and (w,v) and then adding a new arc from the inserted node w to a new leaf) or to a new root (i.e., adding a new root w and new arcs from w to the old root and to a new leaf). The value of Pα,n(∗)(T∗) for T∗∈Tn∗ is determined through all possible ways of constructing cladograms with shape T∗ in this way. More specifically,(1)
ifT1 and T2 denote, respectively, the only cladograms in T1 and T2, let Pα,1′(T1)=Pα,2′(T2)=1;(2)
for everym=3,…,n, let Tm∈Tm be obtained by adding a new leaf labeled m to Tm-1. Then (2)Pα,m′Tm=αm-1-α·Pα,m-1′Tm-1if  the  new  leaf  is  added  to  an  internal  arc  or  to  a  new  root1-αm-1-α·Pα,m-1′Tm-1if  the  new  leaf  is  added  to  a  pendant  arc;(3)
When the desired numbern of leaves is reached, the probability of every tree shape Tn∗∈Tn∗ is defined as (3)Pα,n∗Tn∗=∑πTn=Tn∗Pα,n′Tn.For instance, Figure4 shows the construction of two cladograms in T5 with the same shape and how their probability Pα,5′ is built using the recursion in Step (2). If we generate all cladograms in T5 with this shape, we compute their probabilities Pα,5′, and then we add up all these probabilities, we obtain the probability Pα,5(∗) of this shape, which turns out to be 2(1-α)/(4-α); cf. [4, Figure 23].Figure 4
Two examples of computations of the probabilityPα,n′ of a cladogram through its construction in Step (2) of the definition of the α-model.OncePα,n(∗) is defined on Tn∗, it is transported to Tn, OTn∗, and OTn by defining the probability of an object in one of these sets as the probability of its image in Tn∗ divided by the number of preimages of this image:(i)
For everyT∈Tn, if π(T)=T∗∈Tn∗ and it has k symmetric branch points, then(4)Pα,nT=2kn!·Pα,n∗T∗,becauseπ-1T∗=n!/2k (see, e.g., [4, Lemma 31]).(ii)
For everyTo∈OTn, if πo(To)=T∈Tn, then(5)Pα,noTo=12n-1·Pα,nT,becauseπo-1T=2n-1 (T has 2n-1 different preimages under πo, obtained by taking all possible different combinations of orderings on the n-1 sets child(v), v∈Vint(T∗)).(iii)
For everyTo∗∈OTn∗, if πo,∗(To∗)=T∗∈Tn∗ and it has k symmetric branch points, then(6)Pα,no,∗To∗=12n-k-1·Pα,n∗T∗,becauseπo,∗-1T∗=2n-1-k (from the 2n-1 possible preimages of T∗ under πo,∗, defined by all possible different combinations of orderings on the n-1 sets child(v), v∈Vint(T∗), those differing only on the orderings on the children of the k symmetric branch points are actually the same ordered tree shape).The family(Pα,n(o,∗))n of probabilities of ordered tree shapes satisfies the useful Markov branching recurrence (in the sense of [2, §4]) given by the following proposition. In it and in the sequel, let, for every a,b∈Z+, (7)qαa,b=ΓαaΓαbΓαa+b·φαa,b,where (8)φαa,b=α2a+ba+1-2αa+b-2a-1and Γα:Z+→R is the mapping defined by Γα(1)=1 and, for every n⩾2, Γα(n)=(n-1-α)·Γα(n-1).Proposition 1.
For every0<m<n and for every Tm∗∈OTm∗ and Tn-m∗∈OTn-m∗, (9)Pα,no,∗Tm∗⋆Tn-m∗=qαm,n-mPα,mo,∗Tm∗Pα,n-mo,∗Tn-m∗.This recurrence, together with the fact thatPα,1(o,∗) of a single node is 1, implies that, for every To∗∈OTn∗,(10)Pα,no,∗To∗=∏a,b∈NSTo∗qαa,b.For proofs of Proposition 1 and (10), see Lemma 27 and Proposition 28 in [4], respectively.


## 3. Main Results

Our first result is an explicit formula forPα,n(T), for every n⩾1 and T∈Tn.Proposition 2.
For everyT∈Tn, its probability under the α-model is (11)Pα,nT=2n-1n!·Γαn∏a,b∈NSTφαa,b.Proof.
GivenT∈Tn, let To be any ordered cladogram such that πo(To)=T, and let To∗=π∗(To)∈OTn∗ and T∗=π(T)=πo,∗(To∗). If T∗ has k symmetric branch points, then, by (4), (6), and (10), (12)Pα,nT=2kn!·Pα,n∗T∗=2kn!·2n-k-1·Pα,no,∗To∗=2n-1n!∏a,b∈NSTo∗qαa,b.Now, on the one hand, it is easy to check that (13)NST=min⁡a,b,max⁡a,b∣a,b∈NST0∗,and therefore, since qα is symmetric, (14)Pα,nT=2n-1n!∏a,b∈NSTqαa,b.It remains to simplify this product. If, for every v∈Vint(T), we denote its children by v1 and v2, then (15)∏a,b∈NSTqαa,b=∏v∈VintTΓακTv1ΓακTv2ΓακTvφαNSv.For every v∈Vint(T)∖rT, the term Γα(κT(v)) appears twice in this product: in the denominator of the factor corresponding to v itself and in the numerator of the factor corresponding to its parent. Therefore, all terms Γα(κT(v)) in this product vanish except Γα(κT(rT))=Γα(n) (that appears in the denominator of its factor) and every Γα(κT(v))=Γα(1)=1 with v, a leaf. Thus, (16)Pα,nT=2n-1n!·1Γαn·∏v∈VintTφαNSvas we claimed.Remark 3.
Ford states (see [4, Proposition  32 and page 30]) that if T∈Tn, then (17)Pα,nT=2kn!∏a,b∈NSTq^αa,b,where k is the number of symmetric branching points in T and (18)q^αa,b=2qαa,bifa≠bqαa,bifa=b.If we simplify ∏(a,b)∈NS(T)q^α(a,b) as in the proof of Proposition 2, this formula for Pα,n(T) becomes(19)Pα,nT=2k+mn!·Γαn·∏a,b∈NSTφαa,b,where m is the number of internal nodes whose children have different numbers of descendant leaves. This formula does not agree with the one given in Proposition 2 above, because (20)k+m=n-1-v∈VintT∣childv=v1,v2,κTv1=κTv2butπTv1≠πTv2and, hence, it may happen that k+m<n-1. The first example of a cladogram with this property (and the only one, up to relabeling, with at most 8 leaves) is the cladogram T~∈T8 depicted in Figure 5. For it, our formula gives (see (8.22) in the document ProblsAlpha.pdf in https://github.com/biocom-uib/prob-alpha)(21)Pα,8T~=1-α22-α1267-α6-α5-α3-αwhile expression (19) assigns to T~ a probability of half this value:(22)1-α22-α2527-α6-α5-α3-α.This last value cannot be right, for several reasons. Firstly, by [4, §3.12], when α=1/2, Ford’s model is equivalent to the uniform model, where every cladogram in Tn has the same probability(23)1BTn=12n-3!!and when α=0, Ford’s model gives rise to the Yule model [1, 14], where the probability of every T∈Tn is (24)PYT=2n-1n!∏v∈VintT1κTv-1.In particular, P1/2,8(T~) should be equal to 1/135135 and P0,8(T~) should be equal to 1/19845. Both values are consistent with our formula, while expression (22) yields half these values.
As a second reason, which can be checked using a symbolic computation program, let us mention that if we take expression (22) as the probability of T~ and hence of all other cladograms with its shape, and we assign to all other cladograms in T8 the probabilities computed with Proposition 2, which agree on them with the values given by (19) (they are also provided in the aforementioned document ProblsAlpha.pdf), these probabilities do not add up 1.Figure 5
The cladogramT~∈T8 used in Remark 3.Combining Proposition2 and (4) we obtain the following result.Corollary 4.
For everyT∗∈Tn∗ with k symmetric branch points, (25)Pα,n∗T∗=2n-k-1Γαn∏a,b∈NST∗φαa,b.This formula does not agree, either, with the one given in [4, Proposition 29]: the difference lies again in the same factor of 2 to the power of the number of internal nodes that are not symmetric branch points but whose children have the same number of descendant leaves.The family of density mappings(Pα,n)n satisfies the following Markov branching recurrence.Corollary 5.
For every0<m<n and for every Tm∈Tm and Tn-m∈Tn-m, (26)Pα,nTm⋆Tn-m=2qαm,n-mnmPα,mTmPα,n-mTn-m.Proof.
IfTm∈Tm and Tn-m∈Tn-m, then (27)Pα,mTm=2m-1m!Γαm∏a,b∈NSTmφαa,bPα,n-mTn-m=2n-m-1n-m!Γαn-m∏a,b∈NSTn-mφαa,b,Pα,nTm⋆Tn-m=2n-1n!Γαn∏a,b∈NSTm⋆Tn-mφαa,b=2n-1n!Γαnφαm,n-m∏a,b∈NSTmφαa,b∏a,b∈NSTn-mφαa,b=2n-1n!Γαnφαm,n-mm!Γαm2m-1Pα,mTm·n-m!Γαn-m2n-m-1Pα,n-mTn-m=2qαm,n-mnmPα,mTmPα,n-mTn-mas we claimed.Remark 6.
Against what is stated in [4], (Pα,n(∗))n does not satisfy any Markov branching recurrence; that is, there does not exist any symmetric mapping Q:Z+×Z+→R such that, for every k,l⩾1 and for every Tk∈Tk∗ and Tl∈Tl∗, (28)Pα,k+l∗Tk⋆Tl=Qk,l·Pα,k∗Tk·Pα,l∗Tl.Indeed, let Tm∗,T^m∗∈Tm∗ be any two different tree shapes, both with m leaves and k symmetric branch points, for instance, the tree shapes in T6∗ depicted in Figure 6. Then,(29)Pα,m∗Tm∗=2m-k-1Γαm∏a,b∈NSTm∗φαa,b,Pα,m∗T^m∗=2m-k-1Γαm∏a,b∈NST^m∗φαa,b.In this case, Tm∗⋆Tm∗∈T2m∗ has 2k+1 symmetric branch points and therefore (30)Pα,2m∗Tm∗⋆Tm∗=22m-2k-2Γα2m∏a,b∈NSTm∗⋆Tm∗φαa,b=22m-2k-2Γα2mφαm,m∏a,b∈NSTm∗φαa,b2=22m-2k-2Γα2mφαm,mΓαm2m-k-1Pα,m∗Tm∗2=qαm,mPα,m∗Tm∗Pα,m∗Tm∗while Tm∗⋆T^m∗∈T2m∗ has 2k symmetric branch points and therefore (31)Pα,2m∗Tm∗⋆T^m∗=22m-2k-1Γα2m∏a,b∈NSTm∗⋆T^m∗φαa,b=22m-2k-1Γα2mφαm,m∏a,b∈NSTm∗φαa,b∏a,b∈NST^m∗φαa,b=22m-2k-1Γα2mφαm,m·Γαm2m-k-1Pα,m∗Tm∗·Γαm2m-k-1Pα,m∗T^m∗=2qαm,mPα,m∗Tm∗Pα,m∗T^m∗and qα(m,m)≠2qα(m,m). This shows that there does not exist any well-defined, single real number Q(m,m) such that (32)Pα,2m∗T1,m∗⋆T2,m∗=Qm,m·Pα,m∗T1,m∗·Pα,m∗T2,m∗for every T1,m∗,T2,m∗∈Tm∗.Figure 6
The tree shapes inT6∗ mentioned in Remark 6.



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*Source: 1916094-2018-04-18.xml*