Computing the Distribution of a Tree Metric

Abstract—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. Index Terms—Biology and genetics, discrete mathematics applications, trees, phylogenetics, Robinson-Foulds distance, Poisson approximation, normalization constant. Ç