Parsimony with General Character Evolution

The questions of how to test hypotheses of character evolution, and incorporate character evolution into phylogenetic analysis are considered in this thesis. When a character transformation series is not linear, it is usually decoded into a set of binary characters by various binary coding methods. The disadvantages of such an approach have been discussed extensively in the literature. We propose a new approach together with the corresponding parsimony criteria for working with nonlinear transformation series. In particular, assuming a rooted character state tree is used to model the character evolution, the classical smallphylogenyproblem is extended by given not only the phylogenetic tree but also a character tree. Our techniques are based on finding tree minor embeddings of labeled trees. Three generalizations of tree minor are defined: rooted tree minor, relax-minor andpseudominor. Two new metrics, bag cost and arc cost, are also introduced as the target scoring functions of the problem. The bag cost is analogous to unweighted parsimony while the arc cost is analogous to weighted parsimony, that is, it allows a cost for each state transition. We show that the problem of finding minimum bag cost under relax-minor is NP-hard, however the problems of finding minimum pseudo-minor bag cost and minimum pseudo-minor arc cost can both be solved in linear time. Our algorithm for minimum pseudo-minor arc cost yields the same output for any character tree and phylogenetic tree as Sankoff's algorithm since a character tree can be transformed to a cost matrix. However our algorithm runs in linear time as opposed to the quadratic running time of Sankoff's. One application of our algorithms is the evaluation of multiple phylogenetic trees for a same set of species with the given character trees. Another application would be to test hypotheses for the evolution of a multistate character based on the given phylogenetic tree.