Null Models for the Number of Evolutionary Steps in a Character on a Phylogenetic Tree

Random trees and random characters can be used in null models for testing phylogenetic hypothesis. We consider three interpretations of random trees: first, that trees are selected from the set of all possible trees with equal probability; second, that trees are formed by random speciation or coalescence (equivalent); and third, that trees are formed by a series of random partitions of the taxa. We consider two interpretations of random characters: first, that the number of taxa with each state is held constant, but the states are randomly reshuffled among the taxa; and second, that the probability each taxon is assigned a particular state is constant from one taxon to the next. Under null models representing various combinations of randomizations of trees and characters, exact recursion equations are given to calculate the probability distribution of the number of character state changes required by a phylogenetic tree. Possible applications of these probability distributions are discussed. They can be used, for example, to test for a panmictic population structure within a species or to test phylogenetic inertia in a character's evolution. Whether and how a null model incorporates tree randomness makes little difference to the probability distribution in many but not all circumstances. The null model's sense of character randomness appears more critical. The difficult issue of choosing a null model is discussed.