Distributions of Tree Comparison Metrics--Some New Results

Measures of dissimilarity (metrics) for comparing trees are important tools in the quantitative analysis of evolutionary trees, but many of their properties are incompletely known. The present paper reports formulae for the distributions of three classes of tree comparison metrics: the partition (or symmetric difference) metric, the quartet metric (which compares subsets of four taxa), and a metric based on path-length differences between pairs of taxa. The properties studied include the mean and variance for several underlying distributions of trees, the range, the effect of the number of taxa, and methods of calculation. Three basic theorems and their proofs are reported, one for each class of tree comparison metric. The partition metric generates an asymptotic Poisson distribution for most distributions of trees (its mean is given for three tree distributions). Exact expressions are derived for the variance of the quartet metric and the mean square value of a metric based on path differences. Factors that affect the choice of a metric for a particular study include the degree of similarity of the trees being compared and the type of hypothesis being tested (e.g., whether the trees estimate the same underlying phylogeny or are simply related in some, perhaps unknown, way).