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 for mulae for the distributions of three classes of tree comparison metric s: the partition (or symmetric difference) metric, the quartet metric (which compares subsets of four taxa), and a metric based on path-leng th differences between pairs of taxa. The properties studied include t he mean and variance for several underlying distributions of trees, th e range, the effect of the number of taxa, and methods of calculation. Three basic theorems and their proofs are reported, one for each clas s of tree comparison metric. The partition metric generates an asympto tic 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 met ric based on path differences. Factors that affect the choice of a met ric for a particular study include the degree of similarity of the tre es being compared and the type of hypothesis being tested (e.g., wheth er the trees estimate the same underlying phylogeny or are simply rela ted in some, perhaps unknown, way).