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. Key words: Labeled plane trees, k-ary trees, Dyck paths, bijections Preprint submitted to European Journal of Combinatorics 22 September 2009 

• Type: Journal
• Year: 2010
• Where: EJC
• Authors: Nancy S. S. Gu, Helmut Prodinger, Stephan Wagner
• Keywords: Dyck Paths, EJC 2007, EJC 2010, Labels, Plane Trees