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On a random search tree: asymptotic enumeration of vertices by distance from leaves

Part of: Extremal combinatorics Graph theory Combinatorial probability Combinatorics Lattices

Published online by Cambridge University Press:  08 September 2017

Miklós Bóna  and
Boris Pittel
Miklós Bóna*
Affiliation:
University of Florida
Boris Pittel*
Affiliation:
The Ohio State University
*
* Postal address: Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, FL 32611-8105, USA. Email address: [email protected]
** Postal address: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1175, USA. Email address: [email protected]
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Abstract

A random binary search tree grown from the uniformly random permutation of [n] is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction ck of vertices of a fixed rank k ≥ 0 is shown to decay exponentially with k. We prove that the ranks of the uniformly random, fixed size sample of vertices are asymptotically independent, each having the distribution {ck}. Notoriously hard to compute, the exact fractions ck have been determined for k ≤ 3 only. We present a shortcut enabling us to compute c4 and c5 as well; both are ratios of enormous integers, the denominator of c5 being 274 digits long. Prompted by the data, we prove that, in sharp contrast, the largest prime divisor of the denominator of ck is at most 2k+1 + 1. We conjecture that, in fact, the prime divisors of every denominator for k > 1 form a single interval, from 2 to the largest prime not exceeding 2k+1 + 1.

Keywords

Search treerootleavesrankenumerationasymptoticdistributionnumerical data

MSC classification

Primary: 05A05: Permutations, words, matrices
Secondary: 05A15: Exact enumeration problems, generating functions 05A16: Asymptotic enumeration 05C05: Trees 06B05: Structure theory 05C80: Random graphs 05D40: Probabilistic methods 60C05: Combinatorial probability
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 3 , September 2017 , pp. 850 - 876
Copyright
Copyright © Applied Probability Trust 2017 

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