Abstract
Based on the classic bijective algorithm for trees due 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 the formula for noncrossing trees of size $n$ with $k$ descents and $i$ 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 Hough, which asks for a bijection between two classes of noncrossing trees with a given number of descents.
Highlights
A noncrossing tree (NC-tree for short) is a tree drawn on n points numbered in counterclockwise order on a circle in such a way that its edges are rectilinear and do not cross.We always consider the points labeled counterclockwise from 1 to n and the root labeled1
Based on the classic bijective algorithm for trees due 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
Summary
We derive that the number of labeled NC-trees of size n with k descents and i leaves is n−k n + 1 t t=0 n−k−1 n−k−t n−k 1 n+1−t i k+t−2 n−1−i (n − 1)! For k 1, there is a bijection between labeled NC-tree of size n with k descents and the set Pn,k. In order to show the validity of Step 1, we observe that each merging operation decreases the number of trees without an r-edge by 1 and the number of #-marked vertices by 1.
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