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

Denote by S n the set of all distinct rooted trees with n labeled vertices. A tree is chosen at random in the set S n , assuming that all the possible n n − 1 choices are equally probable. Define τ n ( 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 τ n ( m ) and their asymptotic behavior in the case where m = [ 2 α n ] , 0 < α < ∞ and n → ∞ . In addition, more random trees, branching processes, the Bernoulli excursion and the Brownian excursion are also considered.