Asymptotically normal distribution of some tree families relevant for phylogenetics, and of partitions without singletons

Abstract

P.L. Erdos and L.A. Szekely [Adv. Appl. Math. 10(1989), 488-496] gave a bijection between rooted semilabeled trees and set partitions. L.H. Harper's results [Ann. Math. Stat. 38(1967), 410-414] on the asymptotic normality of the Stirling numbers of the second kind translates into asymptotic normality of rooted semilabeled trees with given number of vertices, when the number of internal vertices varies. The Erdos-Szekely bijection specializes to a bijection between phylogenetic trees and set partitions with classes of size \geq 2. We consider modified Stirling numbers of the second kind that enumerate partitions of a fixed set into a given number of classes of size \geq 2, and obtain their asymptotic normality as the number of classes varies. The Erdos- Szekely bijection translates this result into the asymptotic normality of the number of phylogenetic trees with given number of vertices, when the number of leaves varies. We also obtain asymptotic normality of the number of phylogenetic trees with given number of leaves and varying number of internal vertices, which make more sense to students of phylogeny. By the Erdos-Szekely bijection this means the asymptotic normality of the number of partitions of n + m elements into m classes of size \geq 2, when n is fixed and m varies. The proofs are adaptations of the techniques of L.H. Harper [ibid.]. We provide asymptotics for the relevant expectations and variances with error term O(1/n).