  Efforts to reconstruct phylogenetic trees and understand evolutionary
processes depend fundamentally on stochastic models of speciation and mutation.
The simplest continuous-time model for speciation in phylogenetic trees is the
Yule process, in which new species are "born" from existing lineages at a
constant rate. Recent work has illuminated some of the structural properties of
Yule trees, but it remains mostly unknown how these properties affect sequence
and trait patterns observed at the tips of the phylogenetic tree. Understanding
the interplay between speciation and mutation under simple models of evolution
is essential for deriving valid phylogenetic inference methods and gives
insight into the optimal design of phylogenetic studies. In this work, we
derive the probability distribution of interspecies covariance under Brownian
motion and Ornstein-Uhlenbeck models of phenotypic change on a Yule tree. We
compute the probability distribution of the number of mutations shared between
two randomly chosen taxa in a Yule tree under discrete Markov mutation models.
Our results suggest summary measures of phylogenetic information content,
illuminate the correlation between site patterns in sequences or traits of
related organisms, and provide heuristics for experimental design and
reconstruction of phylogenetic trees.
