Limit theorems for patterns in phylogenetic trees

Abstract

Studying the shape of phylogenetic trees under different random models is an important issue in evolutionary biology. In this paper, we propose a general framework for deriving detailed statistical results for patterns in phylogenetic trees under the Yule-Harding model and the uniform model, two of the most fundamental random models considered in phylogenetics. Our framework will unify several recent studies which were mainly concerned with the mean value and the variance. Moreover, refined statistical results such as central limit theorems, Berry-Esseen bounds, local limit theorems, etc., are obtainable with our approach as well. A key contribution of the current study is that our results are applicable to the whole range of possible sizes of the pattern.

Equality of Shapley value and fair proportion index in phylogenetic trees

“…We first consider the Yule-Harding model [1] which is defined by a tree evolution process: the tree grows by choosing at random one of the leaves and replacing it by a cherry (an internal node with two children). We stop when a tree with n external nodes is constructed.…”

“…We next recall some basic properties of this model; for details see [1]. First, the size of the left subtree I n is uniformly distributed on {1, .…”

“…This definition differs from the standard definition of phylogenetic trees, where one normally assumes that the tree is non-plane and that the leaves are labeled; see for instance Semple and Steel [8]. However, this difference is irrelevant for the scope of the current study, as is frequently the case for probabilistic studies of phylogenetic trees; e.g., see Chang and Fuchs [1].…”

“…These models will be briefly reviewed in Section 3; for more details see [1]. Now, we are going to give definitions of the two parameters above.…”

Repeated fringe subtrees in random rooted trees

“…Fringe subtrees are a classical object of study in the context of random trees, and there is a wealth of results for various random tree models, starting with the fundamental work of Aldous [1]; see [5,6,9,[13][14][15] for recent results on the distribution of fringe subtrees. Fringe subtrees and subtree patterns play an important role, for instance, in the study of phylogenetic trees in mathematical biology [2].…”