  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.
