Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T13:43:37.606Z Has data issue: false hasContentIssue false
  • English
  • Français

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: bona@ufl.edu
** Postal address: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1175, USA. Email address: bgp@math.ohio-state.edu
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aldous, D. (1991). Asymptotic fringe distributions for general families of random trees. Ann. Appl. Prob. 1, 228–266. Google Scholar
[2] Bóna, M. (2014). k-protected vertices in binary search trees. Adv. Appl. Math. 53, 1–11. CrossRefGoogle Scholar
[3] Devroye, L. (1986). A note on the height of binary search trees. J. Assoc. Comput. Mach. 33, 489–498. Google Scholar
[4] Devroye, L. (1991). Limit laws for local counters in random binary search trees. Random Structures Algorithms 2, 303–315. Google Scholar
[5] Devroye, L. and Janson, S. (2014). Protected nodes and fringe subtrees in some random trees. Electron. Commun. Prob. 19, 6. Google Scholar
[6] Drmota, M. (2009). Random Trees: An Interplay Between Combinatorics and Probability. Springer, Vienna. Google Scholar
[7] Du, R. R. X. and Prodinger, H. (2012). Notes on protected nodes in digital search trees. Appl. Math. Lett. 25, 1025–1028. Google Scholar
[8] Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge University Press, Cambridge, UK. Google Scholar
[9] Fuchs, M., Lee, C.-K. and Yu, G.-R. (2016). On 2-protected nodes in random digital trees. Theoret. Comput. Sci. 622, 111–122. Google Scholar
[10] Holmgren, C. and Janson, S. (2015). Asymptotic distribution of two-protected nodes in ternary search trees. Electron J. Prob. 20, 9. Google Scholar
[11] Kesten, H. and Pittel, B. (1996). A local limit theorem for the number of nodes, the height, and the number of final leaves in a critical branching process tree. Random Structures Algorithms 8, 243–299. Google Scholar
[12] Kolchin, V. F. (1978). Moment of degeneration of a branching process and height of a random tree. Math. Notes Acad. Sci. USSR 24, 954–961. Google Scholar
[13] Mahmoud, H. and Pittel, B. (1984). On the most probable shape of a search tree grown from a random permutation. SIAM J. Algebraic Discrete Meth. 5, 69–81. Google Scholar
[14] Mahmoud, H. M. and Ward, M. D. (2012). Asymptotic distribution of two-protected nodes in random binary search trees. Appl. Math. Lett. 25, 2218–2222. Google Scholar
[15] Mahmoud, H. M. and Ward, M. D. (2015). Asymptotic properties of protected notes in random recursive trees. J. Appl. Prob. 52, 290–297. Google Scholar
[16] Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math. 1875). Springer, Berlin. Google Scholar
[17] Pittel, B. (1984). On growing random binary trees. J. Math. Anal. Appl. 103, 461–480. Google Scholar
[18] Pittel, B. (1994). Note on the heights of random recursive trees and random m-ary search trees. Random Structures Algorithms 5, 337–347. Google Scholar