Two involutions on binary trees and generalizations

This paper investigates two involutions on binary trees. One is the mirror symmetry of binary trees which combined with the classical bijection φ 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 φ 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.