On the number of deepest nodes in ordered trees (Q913809)

Using an interesting combinatorial identity for the number of n-node ordered trees with r nodes of maximum level k it is proved that a tree has two deepest nodes on the average, if all n-node trees are regarded equally likely. Furthermore the average number of deepest nodes, if all n-nodes trees with height k are regarded equally likely, is analyzed asymptotically. The reviewer, as well as several of his colleagues, are astonished about the fact that this paper, which was submitted in March 1982, was not published before 1990.