The height distribution of leaves in rooted trees

By the use of the double Hankel contour integration the number of trees of size n (in a simply generated family of trees) such that the mth leaf has height k is asymptotically evaluated. It turns out that the height of the cnth leaf, where c is a constant, is asymptotically Maxwell distributed. This generalizes and extends results by Gutjahr and Pflug [5] and Kirschenhofer [10,12], where completely different methods have been used. In addition the joint height distribution of two different leaves is established. All results can be interpreted in terms of Galton-Watson branching processes.