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The Number of m-ary Search Trees on n Keys

Published online by Cambridge University Press:  01 December 1997

JAMES ALLEN FILL
Affiliation:
Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD 21218–2682, USA; (e-mail: jimfill@jhu.edu)
ROBERT P. DOBROW
Affiliation:
Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD 21218–2682, USA; (e-mail: jimfill@jhu.edu)

Abstract

Problems associated with m-ary trees have been studied by computer scientists and combinatorialists. It is well known that a simple generalization of the Catalan numbers counts the number of m-ary trees on n nodes. In this paper we consider τm, n, the number of m-ary search trees on n keys, a quantity that arises in studying the space of m-ary search trees under the uniform probability model. We prove an exact formula for τm, n, both by analytic and by combinatorial means. We use uniform local approximations for sums of i.i.d. random variables to study the asymptotic development of τm, n for fixed m as n→∞.

Type
Research Article
Copyright
1997 Cambridge University Press

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