The distributions under two species-tree models of the number of root ancestral configurations for matching gene trees and species trees

Abstract

For a pair consisting of a gene tree and a species tree, the ancestral configurations at a species-tree internal node are the distinct sets of gene lineages that can be present at that node. The enumeration of root ancestral configurations—ancestral configurations at the species-tree root—assists in describing the complexity of gene-tree probability calculations in evolutionary biology. Assuming that the gene tree and species tree match in topology, we study the distribution of the number of root ancestral configurations of a random labeled tree topology under the uniform and Yule–Harding models. We employ analytic combinatorics, considering ancestral configurations in the context of additive tree parameters and using singularity analysis to evaluate asymptotic growth of the coefficients of generating functions. For both models, we obtain asymptotic lognormal distributions for the number of root ancestral configurations. For Yule–Harding random trees, we also obtain the asymptotic mean (∼1.425n) and variance (∼2.045n) of the number of root ancestral configurations, paralleling previous results for the uniform model (mean (4/3)n, variance ∼1.822n). A methodological innovation is that to obtain the Yule–Harding asymptotic variance, singularity analysis is conducted from the Riccati differential equation that the generating function satisfies—without possessing the generating function itself.