Involutions on Tip-Augmented Plane Trees for Leaf Interchanging

TL;DR

This work provides bijective proofs for the refined leaf-symmetry $M_n(i, j, k, r, s) = M_n(k, j, i, r, s)$ in tip-augmented plane trees by constructing two involutions. The first, $\Phi$, is defined inductively on unlabelled trees and swaps singleton leaves with elder non-twin leaves while preserving other leaf types; the second, $\Psi$, is defined on labelled trees via Chen's decomposition into matches and a merging procedure, exchanging singleton and elder non-twin leaves without disturbing elder twin, younger, or second leaves. Together, these involutions provide a combinatorial bijective explanation that complements generating-function proofs of the symmetry. The results deepen understanding of refined Narayana and Motzkin-type polynomials for tip-augmented plane trees and illuminate how leaf-type classifications interact with structural decompositions.

Abstract

This paper constructs two involutions on tip-augmented plane trees, as defined by Donaghey, that interchange two distinct types of leaves while preserving all other leaves. These two involutions provide bijective explanations addressing a question posed by Dong, Du, Ji, and Zhang in their work.

Figures (7)

• Figure 1: Involution $\Phi$ on tip-augmented plane trees with $4$ edges
• Figure 2: Tip-augmented plane trees of Type A
• Figure 3: Tip-augmented plane trees of Type B
• Figure 4: Involution $\Phi$ on tip-augmented plane trees with $5$ edges
• Figure 5: An example of involution $\Phi$

Theorems & Definitions (6)

• Theorem 1.1
• Proposition 2.1
• Theorem 3.1
• Remark 1
• Proposition 4.1
• Theorem 4.2