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

A 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 \(k+1\). These trees are known to be related to \((k+1)\)-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 \((2k+1)\)-ary trees.