Bijections for a class of labeled plane trees

Authors:
Nancy S. S. Gu;Helmut Prodinger;Stephan Wagner
Affiliations:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, PR China;Department of Mathematical Sciences, Stellenbosch University, 7602 Stellenbosch, South Africa;Department of Mathematical Sciences, Stellenbosch University, 7602 Stellenbosch, South Africa
Venue:
European Journal of Combinatorics
Year:
2010

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Abstract

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.