  The branching structure of biological evolution confers statistical
dependencies on phenotypic trait values in related organisms. For this reason,
comparative macroevolutionary studies usually begin with an inferred phylogeny
that describes the evolutionary relationships of the organisms of interest. The
probability of the observed trait data can be computed by assuming a model for
trait evolution, such as Brownian motion, over the branches of this fixed tree.
However, the phylogenetic tree itself contributes statistical uncertainty to
estimates of other evolutionary quantities, and many comparative evolutionary
biologists regard the tree as a nuisance parameter. In this paper, we present a
framework for analytically integrating over unknown phylogenetic trees in
comparative evolutionary studies by assuming that the tree arises from a
continuous-time Markov branching model called the Yule process. To do this, we
derive a closed-form expression for the distribution of phylogenetic diversity,
which is the sum of branch lengths connecting a set of taxa. We then present a
generalization of phylogenetic diversity which is equivalent to the expected
trait disparity in a set of taxa whose evolutionary relationships are generated
by a Yule process and whose traits evolve by Brownian motion. We derive
expressions for the distribution of expected trait disparity under a Yule tree.
Given one or more observations of trait disparity in a clade, we perform fast
likelihood-based estimation of the Brownian variance for unresolved clades. Our
method does not require simulation or a fixed phylogenetic tree. We conclude
with a brief example illustrating Brownian rate estimation for thirteen
families in the Mammalian order Carnivora, in which the phylogenetic tree for
each family is unresolved.
