An involution on β(1,0)-trees

In [A. Claesson, S. Kitaev, E. Steingrímsson, Decompositions and statistics for β(1,0)-trees and nonseparable permutations, Adv. in Appl. Math. 42 (2009) 313–328] we introduced an involution, h, on β(1,0)-trees. We neglected, however, to prove that h indeed is an involution. In this note we provide the missing proof. We also refine an equidistribution result given in the same paper.