A Decomposition Algorithm for Noncrossing Trees.

Based on the classic bijective algorithm for trees clue to Chen, we present a decomposition algorithm for noncrossing trees. This leads to a combinatorial interpretation of a formula on noncrossing trees of size n with k descents. We also derive t he formula for noncrossing trees of size n with k descents and leaves, which is a refinement of the formula given by Flajolet and Noy. As an application of our algorithm, we answer a question proposed by 'lough, which asks for a bijection between two classes of noncrossing trees with a given number of descents.