Expected length of pendant and interior edges of a Yule tree

TL;DR

This paper analyzes the expected lengths of edges in a Yule tree, comparing pendant and interior edges, providing formulas and examining how conditioning and parameter estimation affect these expectations.

Contribution

It offers new exact formulas for expected edge lengths in Yule trees and explores the impact of conditioning and parameter estimation on these expectations.

Findings

Expected length of a random edge derived

Expected length of a pendant edge compared

Results depend on conditioning and estimation methods

Abstract

The Yule (pure-birth) model is the simplest null model of speciation; each lineage gives rise to a new lineage independently with the same rate $\lambda$. We investigate the expected length of an edge chosen at random from the resulting evolutionary tree. In particular, we compare the expected length of a randomly selected edge with the expected length of a randomly selected pendant edge. We provide some exact formulae, and show how our results depend slightly on whether the depth of the tree or the number of leaves is conditioned on, and whether $\lambda$ is known or is estimated using maximum likelihood.