  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.
  In the current study, we propose a novel method for measuring the specificity
of a given phylogenetic network in terms of the total number of distributions
of 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 non-binary 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.
