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P.L. Erdos and L.A. Szekely [Adv. Appl. Math. 10(1989), 488-496] gave abijection between rooted semilabeled trees and set partitions. L.H. Harper'sresults [Ann. Math. Stat. 38(1967), 410-414] on the asymptotic normality of theStirling numbers of the second kind translates into asymptotic normality ofrooted semilabeled trees with given number of vertices, when the number ofinternal vertices varies. The Erdos-Szekely bijection specializes to abijection between phylogenetic trees and set partitions with classes of size\geq 2. We consider modified Stirling numbers of the second kind that enumeratepartitions of a fixed set into a given number of classes of size \geq 2, andobtain their asymptotic normality as the number of classes varies. The Erdos-Szekely bijection translates this result into the asymptotic normality of thenumber of phylogenetic trees with given number of vertices, when the number ofleaves varies. We also obtain asymptotic normality of the number ofphylogenetic trees with given number of leaves and varying number of internalvertices, which make more sense to students of phylogeny. By the Erdos-Szekelybijection 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. Theproofs are adaptations of the techniques of L.H. Harper [ibid.]. We provideasymptotics for the relevant expectations and variances with error term O(1/n).

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