Counting Ordered Trees by Permuting Their Parts

operators, n + 1 distinct variables, and n ? 1 pairs of parentheses, how many ways can we combine these symbols into well-formed, explicitly associated expressions? In this article we show that the answer must be (2n)\, because the n-operator ex pressions stand in one-to-one correspondence with permutations of length In. As a corollary, we derive the Catalan numbers. Generalizing, we then enumerate the or dered trees that contain a specified number of nodes of each degree. Finally, we extend the result for trees to ordered forests. These results are not new: my goal is merely to explain them in a new way. 2. EXPRESSIONS AS PERMUTATIONS. In defining a bijective mapping from explicitly associated expressions to permutations, we take as a given that such expres sions are isomorphic to their representations as labeled extended binary trees. Fig ure la shows the expression (y x z) + x as a tree of this kind, to which the edge labels a, b, c, and d have been added. Think of these edge labels as shorthand for left(+), right(+), left(x), and right(x), respectively. The edge labels thus contribute no new information, as they can be inferred from their positions with respect to the operators; conversely, from an edge label we can infer the operator that belongs immediately above it, as well as the edge label's position left-to-right and its sibling edge label.