On the distribution of the number of vertices in layers of random trees (Q1180680)

Summary: 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 \(\tau_ n(m)\) 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 \(\tau_ n(m)\) and their asymptotic behavior in the case where \(m=[2\alpha\sqrt n]\), \(0<\alpha<\infty\) and \(n\to\infty\). In addition, more random trees, branching processes, the Bernoulli excursion and the Brownian excursion are also considered.