Skew Diagrams and Ordered Trees

Abstract

We use a known combinatorial argument to prove that among all ordered trees the ratio of the total number of vertices to leaves is two. We introduce a new combinatorial bijection on the set of these trees that shows why this must be so. Ordered trees are then enumerated by number of leaves, total path length, and number of vertices to obtain q-analogs of Catalan numbers. The results on ordered trees are then readily transferred by the skew diagrams to help enumerate parallelogram polyominoes by their area, perimeter, and other statistics.