Advance Access published on March 7, 2006 doi10.1093comjnlbxk003 Skarbek’s Algorithm for t.pdfVIP

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Advance Access published on March 7, 2006 doi10.1093comjnlbxk003 Skarbek’s Algorithm for t.pdf

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Advance Access published on March 7, 2006 doi10.1093comjnlbxk003 Skarbek’s Algorithm for t

Skarbek’s Algorithm for t-Ary Trees James F. Korsh Department of Computer and Information Sciences, Temple University, Philadelphia, PA, USA Corresponding author: korsh@ This paper presents a new algorithm for generating all n node t-ary trees in linked representation by providing a context and generalizing Skarbek’s algorithm for generating binary trees. It initially appeared that Skarbek’s algorithm could not be generalized to even 3-ary trees, which motivated the author to previously develop an algorithm for generating all n node t-ary trees in linked representation based on a particular listing of integer sequences representing the trees. The context for the generalization of Skarbek’s algorithm turns out to also be a particular listing of integer sequences representing the trees. The paper also gives two constant average time implementations. The first generates the next tree from its predecessor given in linked representation while the second retains additional information about the predecessor beyond just its linked representation. Keywords: combinatorial generation, t-ary trees, constant average time Received 8 October 2005; revised 5 January 2006 1. INTRODUCTION Many algorithms [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] exist for generating binary or t-ary trees represented by sequences of integers but few [28, 29, 30, 31, 32, 33] use linked representation. Only two of them generate the next tree directly from its predecessor. One is the adaptation of Skarbek’s [34] algorithm for generating all n node ordered trees to the generation of n node binary trees by Knuth [32], who showed the adaptation asymptotically required only 8 2/3 10/3n + O(n2) memory references per tree generated, and posed the problem of generalizing it to 3-ary trees. The other is the algorithm by Korsh [33], which was motivated by the absence of any immediate generalization for Skarbek’s algorithm, and is based on a particular li