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

P.L. Erdős and L.A. Székely [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 Erdős-Székely bijection specializes to a bijection between phylogenetic trees and set partitions with classes of size ≥ 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 ≥ 2, and obtain their asymptotic normality as the number of classes varies. The ErdősSzékely 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 Erdős-Székely bijection this means the asymptotic normality of the number of partitions of n+m elements into m classes of size ≥ 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). 1 Semilabeled trees and set partitions Péter Erdős and László Székely [8] enumerated F (n, k), the number of rooted semilabeled trees with k uniquely labeled leaves and n non-root vertices. Such trees have a root, which may or may not have degree one, and is not being counted as vertex or leaf; and have k leaves. Two such trees are identical, if there is a graph isomorphism between them that maps root to root and every leaf label to the same leaf label. The labels of the leaves come from the set {1, 2, . . . , k} and labels are not repeated. Erdős and Székely in [8] established a bijection between the trees counted by F (n, k) and partitions of an n-element set into n − k + 1 classes, under which out-degrees of non-root vertices and the root correspond to class sizes in the partition. The cited result immediately implies that F (n, k) = S(n, n−k+1), where S(a, b) denotes the Stirling number of the second kind that enumerates partitions of an a-element set into b non-empty classes; and