The expected value under the Yule model of the squared path-difference distance

TL;DR

This paper calculates the expected squared path-difference distance between phylogenetic trees under the Yule model, enhancing understanding of its statistical properties for rooted trees.

Contribution

It provides the first known expected value of the squared path-difference under the Yule model for rooted phylogenetic trees.

Findings

Expected value computed for fully resolved rooted trees.

Complements previous results for unrooted trees.

Advances statistical understanding of tree comparison metrics.

Abstract

The path-difference metric is one of the oldest and most popular distances for the comparison of phylogenetic trees, but its statistical properties are still quite unknown. In this paper we compute the expected value under the Yule model of evolution of its square on the space of fully resolved rooted phylogenetic trees with n leaves. This complements previous work by Steel-Penny and Mir-Rossell\'o, who computed this mean value for fully resolved unrooted and rooted phylogenetic trees, respectively, under the uniform distribution.