Denote by Sn the set of all distinct rooted trees with n labeled
vertices. A tree is chosen at random in the set Sn, assuming that all the
possible nn−1 choices are equally probable. Define τn(m) as the number
of vertices in layer m, that is, the number of vertices at a distance m
from the root of the tree. The distance of a vertex from the root is the
number of edges in the path from the vertex to the root. This paper is
concerned with the distribution and the moments of τn(m) and their
asymptotic behavior in the case where m=[2αn], 0<α<∞ and n→∞.
In addition, more random trees, branching processes, the Bernoulli
excursion and the Brownian excursion are also considered.