Denote by Sn the set of all distinct rooted binary trees with n unlabeled vertices. Define σn as a total height of a tree chosen at random in the set S(n), assuming that all the possible choices are equally probable. The total height of a tree is defined as the sum of the heights of its vertices. The height of a vertex in a rooted tree is the distance from the vertex to the root of the tree, that 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 σn and their asymptotic behavior as n → ∞).
