Refined enumeration of $k$-plane trees and $k$-noncrossing trees

TL;DR

This paper introduces a refined enumeration formula for $k$-plane and $k$-noncrossing trees, linking them to generalized Catalan numbers and related tree structures, with several corollaries and analogous theorems.

Contribution

It provides a simple, refined counting formula for $k$-plane and $k$-noncrossing trees, connecting them to known tree families and generalised Catalan numbers.

Findings

Derived a refined counting formula for $k$-plane trees.

Established an analogous theorem for $k$-noncrossing trees.

Connected these trees to $(k+1)$-ary and $(2k+1)$-ary trees.

Abstract

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.