Measuring inconsistency in phylogenetic trees

Suppose that we seek a tree T giving the phylogenetic relationships among the species in a set S. A common method selects for such a tree a maximum parsimony tree using the genome of the species in S. Suppose that K is a proper subset of S. Then T induces a tree U which gives the same relationships among the species in K but omits the species of S which are not in K. Unfortunately, when T is a maximum parsimony tree for the species in S, then U need not be a maximum parsimony tree for the species in K. This phenomenon exhibits an inconsistency in the criterion of maximum parsimony-maximum parsimony trees for different groups of species may be "inconsistent". It implies that the addition of a new species scan change relationships already "established" for prior species if the trees are obtained by the criterion of maximum parsimony. The phenomenon occurs both in artificial examples and with real data. An alternative method for generating phylogenetic trees seeks to minimize such inconsistencies. For each group J consisting of four of the species, we find a tree T(J) describing the relationship only among the four species in J, for example by the use of maximum parsimony on those four species alone. In favorable cases one may combine all the trees T(J) into a single tree T that is consistent with all the trees T(J). If such a tree T exists, then it is unique, and there is a computationally efficient algorithm for finding the tree T. In unfavorable bases such a tree T does not exist, but there may still be a tree containing only "mild" inconsistencies with the trees T(J). A numerical measure is given for the inconsistency I(T) of a tree T in terms of the treelengths of the various trees with set J of leaves in comparison with the tree T. We may then seek a "minimally inconsistent tree T" that minimizes the inconsistency I(T). We describe procedures which find a tree T with low inconsistency I(T). Examples are provided using both artificial strings and data from the complete mitochondrial DNA sequences for 16 species. In particular, minimally inconsistent trees are identified for the 16 species. The definition permits a proof that the trees are in fact minimally inconsistent. The criterion can be applied in both a relative and an absolute sense.