A general approach to proving the minimality of phylogenetic trees illustrated by an example with a set of 23 vertebrates.

We have recently described a method of building phylogenetic trees and have outlined an approach for proving whether a particular tree is optimal for the data used. In this paper we describe in detail the method of establishing lower bounds on the length of a minimal tree by partitioning the data set into subsets. All characters that could be involved in duplications in the data are paired with all other such characters. A matching algorithm is then used to obtain the pairing of characters that reveals the most duplications in the data. This matching may still not account for all nucleotide substitutions on the tree. The structure of the tree is then used to help select subsets of three or more characters until the lower bound found by partitioning is equal to the length of the tree. The tree must then be a minimal tree since no tree can exist with a length less than that of the lower bound. The method is demonstrated using a set of 23 vertebrate cytochrome c sequences with the criterion of minimizing the total number of nucleotide substitutions. There are 131130 7045768798 96033440625 topologically distinct trees that can be constructed from this data set. The method described in this paper does identify 144 minimal tree variants. The method is general in the sense that it can be used for other data and other criteria of length. It need not however always be possible to prove a treee minimal but the method will give an upper and lower bound on the length of minimal trees.