On the number of deepest nodes in ordered trees

AbstractLet Qn,k,r be the number of all n-node ordered trees with r nodes of maximum level k and let Bn,k,r be the number of all r-tuply rooted ordered trees with n nodes and height less than or equal to k. In this paper we derive the identitity Qn,k,r = Bn+1,k,r+1 − Bn+1,k,r + Bn,k,r−1 where n, k, r>0. An explicit expression for Qn,k,r and its asymptotic equivalent is computed. Assuming that all trees with n nodes are equally likely, the above relation implies that a tree has two deepest nodes on the average; the probabilities and higher moments about origin for this distribution are computed. Finally, assuming that all n-node trees with height k are equally likely, we show that such a tree has 4n+1k+1sin2πk+1-6k+1+O1n deepest nodes on the average for fixed k