Enumeration of binary trees compatible with a perfect phylogeny

Evolutionary models used for describing molecular sequence variation suppose that at a non-recombining genomic segment, sequences share ancestry that can be represented as a genealogy—a rooted, binary, timed tree, with tips corresponding to individual sequences. Under the infinitely-many-sites mutation model, mutations are randomly superimposed along the branches of the genealogy, so that every mutation occurs at a chromosomal site that has not previously mutated; if a mutation occurs at an interior branch, then all individuals descending from that branch carry the mutation. The implication is that observed patterns of molecular variation from this model impose combinatorial constraints on the hidden state space of genealogies. In particular, observed molecular variation can be represented in the form of a perfect phylogeny, a tree structure that fully encodes the mutational differences among sequences. For a sample of n sequences, a perfect phylogeny might not possess n distinct leaves, and hence might be compatible with many possible binary tree structures that could describe the evolutionary relationships among the n sequences. Here, we investigate enumerative properties of the set of binary ranked and unranked tree shapes that are compatible with a perfect phylogeny, and hence, the binary ranked and unranked tree shapes conditioned on an observed pattern of mutations under the infinitely-many-sites mutation model. We provide a recursive enumeration of these shapes. We consider both perfect phylogenies that can be represented as binary and those that are multifurcating. The results have implications for computational aspects of the statistical inference of evolutionary parameters that underlie sets of molecular sequences.