Bijections of k -plane trees

A k -plane tree is a tree drawn in the plane such that the vertices are labeled by integers in the set { 1 , 2 , … , k } , the children of all vertices are ordered, and if ( i , j ) is an edge in the tree, where i and j are labels of adjacent vertices in the tree, then i + j ≤ k + 1 . In this paper, we construct bijections between these trees and the sets of k -noncrossing increasing trees, locally oriented ( k − 1 ) -noncrossing trees, Dyck paths, and some restricted lattice paths.