On defining a unique phylogenetic tree with homoplastic characters.

This paper discusses the problem of whether creating a matrix with all the character state combinations that have a fixed number of steps (or extra steps) on a given tree T, produces the same tree T when analyzed with maximum parsimony or maximum likelihood. Exhaustive enumeration of cases up to 20 taxa for binary characters, and up to 12 taxa for 4-state characters, shows that the same tree is recovered (as unique most likely or most parsimonious tree) as long as the number of extra steps is within 1/4 of the number of taxa. This dependence, 1/4 of the number of taxa, is discussed with a general argumentation, in terms of the spread of the character changes on the tree used to select character state distributions. The present finding allows creating matrices which have as much homoplasy as possible for the most parsimonious or likely tree to be predictable, and examination of these matrices with hill-climbing search algorithms provides additional evidence on the (lack of a) necessary relationship between homoplasy and the ability of search methods to find optimal trees.