A probability measure for character compatibility

Abstract

It is proved that two undirected binary cladistic characters are compatible iff their smaller states are disjoint or one is a subset of the other. The concept of a cladistic character as an ordered tree of subsets is defined. Cladistic characters that have the same number of elements in their corresponding states are defined to be “nesting equivalent.” The equivalence classes of this relation are called “nestings.” A certain class of n-tuples is shown to have a biunique correspondence with the n!-membered set of all nestings of n binary characters. The model of randomness proposed is that all characters that are nesting equivalent are equally likely. The probability that a pair of undirected binary characters is compatible is derived under this model. This result is extended to collections of undirected binary characters, to collections of directed binary characters, and finally to collections that may include multistate characters. Some proofs are presented which allow a more efficient use of the n-tuple representation of ordered trees of subsets.