title: Statistical properties of pairwise distances between leaves on a random Yule tree.
abstract: A Yule tree is the result of a branching process with constant birth and death rates. Such a process serves as an instructive null model of many empirical systems, for instance, the evolution of species leading to a phylogenetic tree. However, often in phylogeny the only available information is the pairwise distances between a small fraction of extant species representing the leaves of the tree. In this article we study statistical properties of the pairwise distances in a Yule tree. Using a method based on a recursion, we derive an exact, analytic and compact formula for the expected number of pairs separated by a certain time distance. This number turns out to follow a increasing exponential function. This property of a Yule tree can serve as a simple test for empirical data to be well described by a Yule process. We further use this recursive method to calculate the expected number of the n-most closely related pairs of leaves and the number of cherries separated by a certain time distance. To make our results more useful for realistic scenarios, we explicitly take into account that the leaves of a tree may be incompletely sampled and derive a criterion for poorly sampled phylogenies. We show that our result can account for empirical data, using two families of birds species.
authors.affiliation: Max Planck Institute for Molecular Genetics, Berlin, Germany.
authors.forename: Michael
authors.initials: M
authors.lastname: Sheinman
authors.affiliation: Max Planck Institute for Molecular Genetics, Berlin, Germany; INRA, UR1077 Unite Mathematique Informatique et Genome, Jouy-en-Josas, France.
authors.forename: Florian
authors.initials: F
authors.lastname: Massip
authors.affiliation: Max Planck Institute for Molecular Genetics, Berlin, Germany.
authors.forename: Peter F
authors.initials: PF
authors.lastname: Arndt
mesh_terms: D018515:Plant Leaves; D014197:Trees / Q000254:growth & development*
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