Bijections for a class of labeled plane trees

We consider plane trees whose vertices are given labels from the set { 1 , 2 , … , k } in such a way that the sum of the labels along any edge is at most k + 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 ) -ary trees as well as generalized Dyck paths whose step sizes are k (up) and 1 (down) respectively, thereby extending some classic results.