The expected value of the squared euclidean cophenetic metric under the Yule and the uniform models

TL;DR

This paper calculates the expected squared Euclidean cophenetic metric for fully resolved rooted phylogenetic trees with n leaves under Yule and uniform models, aiding in tree comparison analysis.

Contribution

It provides the first explicit formulas for the expected value of the squared cophenetic metric under these common evolutionary models.

Findings

Expected value formulas for Yule and uniform models

Enhances understanding of phylogenetic tree distances

Supports statistical comparison of phylogenetic trees

Abstract

The cophenetic metrics $d_{\varphi,p}$, for $p\in {0}\cup[1,\infty[$, are a recent addition to the kit of available distances for the comparison of phylogenetic trees. Based on a fifty years old idea of Sokal and Rohlf, these metrics compare phylogenetic trees on a same set of taxa by encoding them by means of their vectors of cophenetic values of pairs of taxa and depths of single taxa, and then computing the $L^p$ norm of the difference of the corresponding vectors. In this paper we compute the expected value of the square of $d_{\varphi,2}$ on the space of fully resolved rooted phylogenetic trees with $n$ leaves, under the Yule and the uniform probability distributions.