Binary trees having a given number of nodes with 0 , 1 , and 2 children

Note that k must always equal i+ 1 in a binary tree. Prodinger [P] recently computed the probability that a random binary tree with n nodes has i nodes with 2 children (and hence i + 1 nodes without children and n − 2i − 1 nodes with 1 child). Since the total number of binary trees with n nodes is known—it is bn—, his formulas can be derived easily from the above theorem and vice versa. Prodinger proved his results by simplifying sums of expressions involving binomial coefficients that were derived by Mahmoud [M].