Counting and Randomly Generating Binary Trees

The number b, of binary trees with n nodes is well known. Two ways to determine b, are presented in [3]. One, based on constructing them, leads to recurrence relations solved by generating functions. The other recognizes that b, is the same as the number of stack permutations and counts these permutations. In [l], a method based on well-formed sequences of parentheses is given, and the result is applied to generating a random binary tree in linear time using integers of size no larger than 2n. Enumeration of binary trees is done in [2,5-71. In this paper a simple direct approach is used to find b,. It aIso provides another way to generate a random binary tree as efficiently as in [l]. In [41 this approach gives ranking and unranking algorithms as efficient as those in [7], and allows a more efficient algorithm to determine the next binary tree. For any n > 0, a binary tree with n nodes can be uniquely represented by a sequence of n pairs of O’s and l’s, one pair for each node. In a node’s pair, x y, x (y> specifies whether the node’s left (right) subtree is null or non-null: 0 for null and 1 for non-null. The pairs appear in the order in