Grand Motzkin paths and $\{0,1,2\}$-trees -- a simple bijection

TL;DR

This paper extends a classic bijection between Motzkin paths and certain trees to a broader class called Grand Motzkin paths, linking them to ordered lists of trees with an odd number of components, providing a new perspective.

Contribution

It introduces a novel bijection connecting Grand Motzkin paths with ordered lists of {0,1,2}-trees, expanding the combinatorial understanding beyond existing models.

Findings

Establishes a bijection between Grand Motzkin paths and lists of {0,1,2}-trees.

Provides an alternative framework to recent related work by Rocha and Pereira Spreafico.

Simplifies the combinatorial interpretation of these paths and trees.

Abstract

A well-known bijection between Motzkin paths and ordered trees with outdegree always $\le2$, is lifted to Grand Motzkin paths (the nonnegativity is dropped) and an ordered list of an odd number of such $\{0,1,2\}$ trees. This offers an alternative to a recent paper by Rocha and Pereira Spreafico.