New equidistributions on plane trees and decompositions of $132$-avoiding permutations

Our main results in this paper are new equidistributions on plane trees and $132$-avoiding permutations, two closely related objects. As for the former, we discover a characteristic for vertices of plane trees that is equally distributed as the height for vertices. The latter is concerned with four distinct ways of decomposing a $132$-avoiding permutation into subsequences. We show combinatorially that the subsequence length distributions of the four decompositions are mutually equivalent, and there is a way to group the four into two groups such that each group is symmetric and the joint length distribution of one group is the same as that of the other. Some consequences are discussed. For instance, we provide a new refinement of the equidistribution of internal vertices and leaves, and present new sets of $132$-avoiding permutations that are counted by the Motzkin numbers and their refinements.