On the distribution of the number of vertices in layers of random trees

Denote by the set of all distinct rooted trees with n labeled vertices. A tree is chosen at random in the set , assuming that all the possible choices are equally probable. Define 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 and their asymptotic behavior in the case where , and . In addition, more random trees, branching processes, the Bernoulli excursion and the Brownian excursion are also considered.