Refined Enumeration of $${{\varvec{k}}}$$-plane Trees and $${\varvec{k}}$$-noncrossing Trees

Abstract

AbstractA k-plane tree is a plane tree whose vertices are assigned labels between 1 and k in such a way that the sum of the labels along any edge is no greater than $$k+1$$ k + 1 . These trees are known to be related to $$(k+1)$$ ( k + 1 ) -ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for k-noncrossing trees, a similarly defined family of labelled noncrossing trees that are related to $$(2k+1)$$ ( 2 k + 1 ) -ary trees.