Combinatorial Scoring of Phylogenetic Trees and Networks Based on Homoplasy-Free Characters

Construction of phylogenetic trees and networks for extant species from their characters represents one of the key problems in phylogenomics. While solution to this problem is not always uniquely defined and there exist multiple methods for tree/network construction, it becomes important to measure how well the constructed networks capture the given character relationship across the species. We propose a novel method for measuring the specificity of a given phylogenetic network in terms of the total number of distributions of homoplasy-free character states at the leaves that the network may impose. While for binary phylogenetic trees, this number has an exact formula and depends only on the number of leaves and character states but not on the tree topology, the situation is much more complicated for nonbinary trees or networks. Nevertheless, we develop an algorithm for combinatorial enumeration of such distributions, which is applicable for arbitrary trees and networks under some reasonable assumptions. We further extend our algorithm to a special class of characters that follow Dollo's law of irreversibility.