Bijection for ternary trees and non-crossing trees

Authors:
Alios Panholzer;Helmut Prodinger
Affiliations:
Institut für Algebra und Computermathematik, Technische Universität Wien, Wieder Hauptstrasse 8-10, A-1040 Wein, Austria;School of Mathematics, The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, P.O. Wits, 2050 Johannesburg, South Africa
Venue:
Discrete Mathematics
Year:
2002

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Abstract

The number Nn of non-crossing trees of size n satisfies Nn+1 = Tn, where Tn enumerates ternary trees of size n. We constuct a new bijection to establish that fact. Since Tn = (1/(2n + 1))(3nn), it follows that 3(3n - 1)(3n - 2)Tn-1 =2n(2n + 1)Tn. We construct two bijections "explaining" this recursion; one of them easily extends to the case of t-ary trees.