The Robinson-Foulds (RF) distance is by far the most widely used measure of dissimilarity between trees. Although the distribution of these distances has been investigated for 20 years, an algorithm that is explicitly polynomial time has yet to be described for computing the distribution for trees around a given tree. In this paper, we derive a polynomial-time algorithm for this distribution. We show how the distribution can be approximated by a Poisson distribution determined by the proportion of leaves that lie in "cherries" of the given tree. We also describe how our results can be used to derive normalization constants that are required in a recently proposed maximum likelihood approach to supertree construction. Ç T REE comparison metrics are widely used in phylogenetics for comparing evolutionary trees [3], [9] and performing statistical tests-for example, to test whether two trees are more "significantly different" from each other than one might expect if one or both trees were randomly chosen [6], [7] . In order to address these statistical questions, one needs to determine the distribution of the metric under some null model (see, for example, [6], [7] ). The symmetric difference or Robinson-Foulds metric is the most widely used measure of differences between phylogenetic trees, and its distribution is particularly attractive to study. In a landmark paper [4], the authors described this distribution of trees relative to a fixed reference tree via a system of generating functions. This allowed the authors to calculate the distribution explicitly for small trees and provided a tool for analytic results on this distribution in later work by others. However, the approach described in [4] does not immediately appear to provide a polynomial-time algorithm for computing this distribution, and for larger trees, their approach may be computationally prohibitive. In this paper, we describe how to calculate the distribution of the Robinson-Foulds metric relative to a fixed tree. We also show how the distribution can be approximated by a Poisson distribution whose parameter depends on just one aspect of tree shape-the number of "cherries." Our investigation into the distribution of the metric has also been motivated by its relevance to a recent approach for "supertree" construction that is based on maximum likelihood [12] . In particular, our algorithm allows the normalization constants in the likelihood calculations to be computed explicitly. We describe how these normalization constants depend weakly on aspects of the shape of the tree-for example, how many "cherries" the tree has. We start by recalling some terminology.
Moreover, the distribution of the Robinson-Foulds distance relative to a fixed tree can be computed in linear time REF .
Computing the Distribution of a Tree Metric
IEEE/ACM Transactions on Computational Biology and Bioinformatics
IEEE/ACM Transactions on Computational Biology and Bioinformatics
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