Pleine trees and reduction rules

Abstract

Inspired by the results of Gelfand, Graev and Postnikov (1997) on the reduction rule for binary trees, we present several bijections on different types of reduction rules for plane trees such as symmetric k-ary trees, locally symmetric k-ary trees, symmetric even trees, symmetric noncrossing trees and Motzkin trees. We give the corresponding enumeration results, and prove the equidistribution of some classes of trees.