Bijections for a class of labeled plane trees

We consider plane trees whose vertices are given labels from the set {1,2,…,k}{1,2,…,k} in such a way that the sum of the labels along any edge is at most k+1k+1; it turns out that the enumeration of these trees leads to a generalization of the Catalan numbers. We also provide bijections between this class of trees and (k+1)(k+1)-ary trees as well as generalized Dyck paths whose step sizes are kk (up) and 11 (down) respectively, thereby extending some classic results.