Determining the asymptotic number of phylogenetic trees

Abstract

The phylogenetic (evolutionary) trees of biology are a special class of labelled trees. In graph-theoretic terms, a phylogenetic tree is a tree whose points have been labelled with disjoint subsets of the labelling set. Every point of degree less than three must have a nonempty label. Formulas are found for the exact numbers of phylogenetic trees with n labels, and numerical results are presented for selected n⩽40. The asymptotic behaviour of these numbers as n → ∞ is determined. Similar results are obtained for the mean and variance of the number of points in a phylogenetic tree with n labels. The effect of requiring that each nonempty label be a singleton is also studied.

The second author is grateful for the support of the Australian Research Grants Committee for the project "Numerical Implementation of Unlabeled Graph Counting Algorithms", under which research and computing for this paper were performed.