Statistical Properties of Pairwise Distances between Leaves on a Random Yule Tree

TL;DR

This paper analyzes the statistical properties of pairwise distances in Yule trees, providing exact formulas and methods to test if empirical phylogenetic data fit the Yule model, with applications to bird species data.

Contribution

It derives an exact recursive formula for pairwise distances in Yule trees, including incomplete sampling scenarios, advancing methods for phylogenetic analysis.

Findings

Expected pairwise distances follow an exponential growth pattern.

The recursive method accurately predicts the number of closely related pairs.

The model accounts for incomplete sampling in empirical data.

Abstract

A Yule tree is the result of a branching process with constant birth and death rates. Such a process serves as an instructive null model of many empirical systems, for instance, the evolution of species leading to a phylogenetic tree. However, often in phylogeny the only available information is the pairwise distances between a small fraction of extant species representing the leaves of the tree. In this article we study statistical properties of the pairwise distances in a Yule tree. Using a method based on a recursion, we derive an exact, analytic and compact formula for the expected number of pairs separated by a certain time distance. This number turns out to follow a increasing exponential function. This property of a Yule tree can serve as a simple test for empirical data to be well described by a Yule process. We further use this recursive method to calculate the expected number of the $n$-most closely related pairs of leaves and the number of cherries separated by a certain time distance. To make our results more useful for realistic scenarios, we explicitly take into account that the leaves of a tree may be incompletely sampled and derive a criterion for poorly sampled phylogenies. We show that our result can account for empirical data, using two families of birds species.