  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).
