Enumerating Phylogenetic Trees with Multiple Labels

Abstract

Evolutionary trees are used in biology to illustrate postulated ancestral relationships between species and are often called phylogenetic trees. They can be characterized in graph theoretic terms by certain classes of labelled trees. Disjoint subsets of the labelling set are assigned to tree vertices so that all pendant vertices and any vertices of degree two are labelled. Here we determine exact and asymptotic numbers for two classes of trees in which multiple vertex labels are allowed. In the first class vertices of degree two are forbidden and in the second class vertices of degree greater than two cannot be labelled. A general method is presented for deriving the asymptotic analysis of any multiple label case. Asymptotic results for the two classes of trees under study are then obtained by applying this method to previously published results. This paper completes work by the authors on the enumeration of various classes of phylogenetic trees.