The Probabilities of Trees and Cladograms under Ford's α-Model

Ford's α-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.


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][3][4][5][6][7][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 -model for rooted bifurcating phylogenetic trees or cladograms [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 to pendant arcs, i.e., to arcs ending in leaves) by allocating a possibly different probability (that depends on a parameter and hence its name, " -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 of tree shapes (unlabeled rooted bifurcating trees) are wrong, failing for some trees with ⩾ 8 leaves; see Propositions 29 and 32 in [4], with the definition of̂given 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 page https://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 T of cladograms with leaves labeled 1, . . . , , for every from 2 to 8.

Preliminaries
2.1. Definitions, Notations, and Conventions. Throughout this paper, by a tree , we mean a rooted bifurcating tree. As it is customary, we understand as a directed graph, with its arcs pointing away from the root, which we shall denote by . Then, all nodes in have out-degree either 0 (its leaves, which form the set ( )) or 2 (its internal nodes, which form the set int ( )). The children of an internal node V are those  nodes such that (V, ) is an arc in , and they form the set child(V). A node is a descendant of a node V when there exists a directed path from V to in . For every node V, the subtree V of rooted at V is the subgraph of induced on the set of descendants of V.
A tree is ordered when it is endowed with an ordering ≺ V on every set child(V). A cladogram (resp., an ordered 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 term tree 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. Proposition 1).
An isomorphism of ordered trees is an isomorphism of rooted trees that moreover preserves these orderings. An isomorphism 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, , , are the same, in symbols = , when they are (only) isomorphic. We shall denote by T * and OT * , respectively, the sets of tree shapes and of ordered tree shapes with leaves. Given any finite set of taxa Σ, we shall denote by T Σ and OT Σ , 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 T and OT (with = |Σ|) instead of T Σ and OT Σ , and then we shall understand that Σ is [ ] = {1, 2, . . . , }.
There exist natural isomorphism-preserving forgetful mappings that "forget" the orderings or the labels of the trees. In particular, we shall call the image of a cladogram under its shape. Figure 1 depicts an example of images under these forgetful mappings.
Let us introduce some more notations. For every node V in a tree , (V) is its number of descendant leaves. For every internal node V in an ordered tree , with children Figure 2, then A symmetric branch point in a tree is an internal node V such that if V 1 and V 2 are its children, then the subtrees V 1 and V 2 of 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 cladograms and on Σ and Σ , respectively, with Σ ∩ Σ = 0, their root join is the cladogram ⋆ on Σ∪Σ obtained by connecting the roots of and to a (new) common root ; see Figure 3. If , are ordered cladograms, ⋆ is ordered by inheriting the orderings on and and ordering the children of the new root as ≺ . If and are tree shapes, a similar construction yields a tree shape ⋆ ; if they are moreover ordered, then ⋆ becomes an ordered tree shape as explained above.

The -Model.
Ford's -model [4] defines, for every ⩾ 1, a family of probability density functions ( * ) , on T * that depends on one parameter ∈ [0, 1], and then it translates this family into three other families of probability density functions , on T , ( , * ) , on OT * , and ( ) , on OT , by imposing that the probability of a tree shape is equally distributed among its preimages under , , * , and ∘ = , * ∘ * , respectively. It is well known [13] that every ∈ T can be obtained in a unique way by adding recurrently to a single node labeled 1 new leaves labeled 2, . . . , to arcs (i.e., splitting an arc ( , V) into two arcs ( , ) and ( , V) and then adding a new arc from the inserted node to a new leaf) or to a new root (i.e., adding a new root and new arcs from to the old root and to a new leaf). The value of ( * ) , ( * ) for * ∈ T * is determined through all possible ways of constructing cladograms with shape * in this way. More specifically, (1) if 1 and 2 denote, respectively, the only cladograms in T 1 and T 2 , let ,1 ( 1 ) = ,2 ( 2 ) = 1; (2) for every = 3, . . . , , let ∈ T be obtained by adding a new leaf labeled to −1 . Then if the new leaf is added to an internal arc or to a new root if the new leaf is added to a pendant arc; (2) (3) When the desired number of leaves is reached, the probability of every tree shape * ∈ T * is defined as For instance, Figure 4 shows the construction of two cladograms in T 5 with the same shape and how their probability ,5 is built using the recursion in Step (2). If we generate all cladograms in T 5 with this shape, we compute their probabilities ,5 , and then we add up all these probabilities, we obtain the probability ( * ) ,5 of this shape, which turns out to be 2(1 − )/(4 − ); cf. [4, Figure 23].
Once ( * ) , is defined on T * , it is transported to T , OT * , and OT by defining the probability of an object in one of these sets as the probability of its image in T * divided by the number of preimages of this image: (i) For every ∈ T , if ( ) = * ∈ T * and it has symmetric branch points, then where ( , ) = 2 ( and Γ : Z + → R is the mapping defined by Γ (1) = 1 and, for every ⩾ 2, Γ ( ) = ( − 1 − ) ⋅ Γ ( − 1).

Main Results
Our first result is an explicit formula for , ( ), for every ⩾ 1 and ∈ T .

Proposition 2. For every ∈ T , its probability under the -model is
Proof. Given ∈ T , let be any ordered cladogram such that ( ) = , and let * = * ( ) ∈ OT * and * = ( ) = , * ( * ). If * has symmetric branch points, then, by (4), (6), and (10), Now, on the one hand, it is easy to check that and therefore, since is symmetric, It remains to simplify this product. If, for every V ∈ int ( ), we denote its children by V 1 and V 2 , then For every V ∈ int ( ) \ { }, the term Γ ( (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 Γ ( (V)) in this product vanish except Γ ( ( )) = Γ ( ) (that appears in the denominator of its factor) and every Γ ( (V)) = Γ (1) = 1 with V, a leaf. Thus, as we claimed.
where is the number of symmetric branching points in and̂( If we simplify ∏ ( , )∈NS( )̂( , ) as in the proof of Proposition 2, this formula for , ( ) becomes where 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 and, hence, it may happen that + < − 1. The first example of a cladogram with this property (and the only one, up to relabeling, with at most 8 leaves) is the cladogram̃∈ T 8 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) while expression (19) assigns tõa probability of half this value:   and when = 0, Ford's model gives rise to the Yule model [1,14], where the probability of every ∈ T is In particular, 1/2,8 (̃) should be equal to 1/135135 and 0,8 (̃) 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 of̃and hence of all other cladograms with its shape, and we assign to all other cladograms in T 8 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.