Labeled Plane Trees and Increasing Plane Trees

TL;DR

This work establishes a polynomial analogue of the classical identity $(n+1)!C_n=2^n(2n-1)!!$ by linking labeled plane trees, increasing plane trees, and Stirling permutations via improper edges. A new edge-wise involution on labeled plane trees reverses the proper/improper status of a chosen edge while preserving all other structure, enabling a weight-preserving bijection between labeled plane trees and increasing plane trees when edges are appropriately labeled. Consequently, the authors derive exact polynomial identities: $P_n(x,y)=(2n-1)!!(x+y)^n$ and $O_n(x,y,t)=\sum_{r=1}^n t^r S_{n,r}(x+y)^{n-r}$, where $S_{n,r}$ counts increasing plane trees with root degree $r$ (equivalently Stirling permutations with $r$ blocks). Generating-function relations and a depth-first walk bijection to Stirling permutations further illuminate the structure, connecting combinatorial interpretations with polynomial encodings and enriching the understanding of edge classifications in plane trees.

Abstract

This note is dedicated to presenting a polynomial analogue of $(n+1)!C_n=2^n(2n-1)!!$ (with $C_n$ as the $n$-th Catalan number) in the context of labeled plane trees and increasing plane trees, based on the definition of improper edges in labeled plane trees. A new involution on labeled plane trees is constructed to establish this identity, implying that the number of improper edges and the number of proper edges are equidsitributed over the set of labeled plane trees.

Figures (5)

• Figure 1: A labeled plane tree with $7$ edges.
• Figure 2: An increasing plane tree with $7$ edges.
• Figure 3: The decomposition of a labeled plane tree $T$ based on edge $e$
• Figure 5: An example of the bijection $\Phi=\phi_{e_1}\phi_{e_2}\phi_{e_3}$
• Figure : $T$

Theorems & Definitions (4)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2