Two involutions on binary trees and generalizations

TL;DR

This paper explores two involutions on binary trees, generalizing them to broader classes, and reveals new combinatorial distributions and equidistributions related to plane and rooted labeled trees.

Contribution

Introduces two involutions on binary trees, generalizes them to weakly increasing and rooted labeled trees, and establishes new equidistributions involving Catalan's triangle.

Findings

Mirror involution combined with classical bijection solves an open problem.

Generalized involutions lead to equidistributions on weakly increasing trees.

New statistic on plane trees follows the distribution of Catalan's triangle.

Abstract

This paper investigates two involutions on binary trees. One is the mirror symmetry of binary trees which combined with the classical bijection $\varphi$ between binary trees and plane trees answers an open problem posed by Bai and Chen. This involution can be generalized to weakly increasing trees, which admits to merge two recent equidistributions found by Bai--Chen and Chen--Fu, respectively. The other one is constructed to answer a bijective problem on di-sk trees asked by Fu--Lin--Wang and can be generalized naturally to rooted labeled trees. This second involution combined with $\varphi$ leads to a new statistic on plane trees whose distribution gives the Catalan's triangle. Moreover, a quadruple equidistribution on plane trees involving this new statistic is proved via a recursive bijection.