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

Two involutions on binary trees and generalizations