On asymptotic joint distributions of cherries and pitchforks for random phylogenetic trees

Abstract

Tree shape statistics provide valuable quantitative insights into evolutionary mechanisms underpinning phylogenetic trees, a commonly used graph representation of evolutionary relationships among taxonomic units ranging from viruses to species. We study two subtree counting statistics, the number of cherries and the number of pitchforks, for random phylogenetic trees generated by two widely used null tree models: the proportional to distinguishable arrangements (PDA) and the Yule-Harding-Kingman (YHK) models. By developing limit theorems for a version of extended Pólya urn models in which negative entries are permitted for their replacement matrices, we deduce the strong laws of large numbers and the central limit theorems for the joint distributions of these two counting statistics for the PDA and the YHK models. Our results indicate that the limiting behaviour of these two statistics, when appropriately scaled using the number of leaves in the underlying trees, is independent of the initial tree used in the tree generating process.

Limit Theorems for Patterns in Ranked Tree-Child Networks

“…Note that the second, third and fourth term on the right hand side of ( 15) and all terms in R n are of a smaller lexicographic order than (s, r, t). Also note that (15) has the form (6). Now, first the induction base holds because of (11).…”

“…Studying properties of shape statistics for random models that are used to describe the evolutionary relationship between species is an important topic in biology. For phylogenetic trees, which are used to model non-reticulate evolution, many such studies have been performed and the stochastic behavior of, e.g., pattern counts are known in great detail; see [4,6,7,8,14,15,17,19,20]. On the other hand, for phylogenetic networks, which are used to model reticulate evolution, very little is known about the number of occurrences of patterns when the networks from a given class are randomly sampled.…”

“…In order to prove the claim for (s, r, t), we observe that (19) has the form (6) with the sequence ψ n satisfying ψ n ∼ cr,s,t n (r+s+t)/2−1 where cr,s,t :=4sc…”

On a mathematical connection between single-elimination sports tournaments and evolutionary trees

“…It is often convenient for evolutionary models to assume that each sequence of branching events that could produce a rooted binary tree for a set of labeled species is equally likely; this assumption, that of the Yule or Yule-Harding model in phylogenetics [1,13,15,20,27,28,30,33,35], produces a uniform distribution on labeled histories. Computations concerning features of tree shape for evolutionary trees often evaluate the probability that such features are produced under the Yule-Harding model, so that they directly or indirectly examine the fraction of labeled histories on y leaves that possess a given feature, or the probability distribution of a quantity across labeled histories [3,5,6,7,8,10,12,16,18,22,24,25,29,31,34,36,37]. Mathematical phylogenetics computations have used combinatorial results on the set of labeled histories for y species, for example employing a space of labeled histories with a notion of distance between them [26] and a characterization of the labeled topologies that possess the largest number of labeled histories [9,11,14].…”

“…Computations concerning features of tree shape for evolutionary trees often evaluate the probability that such features are produced under the Yule–Harding model, so that they directly or indirectly examine the fraction of labeled histories on y leaves that possess a given feature, or the probability distribution of a quantity across labeled histories [3, 5, 6, 7, 8, 10, 12, 16, 18, 22, 24, 25, 29, 31, 34, 36, 37]. Mathematical phylogenetics computations have used combinatorial results on the set of labeled histories for y species, for example employing a space of labeled histories with a notion of distance between them [26] and a characterization of the labeled topologies that possess the largest number of labeled histories [9, 11, 14].…”