Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-25T23:39:25.843Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Subtree Size Profile of Binary Search trees

Published online by Cambridge University Press:  22 January 2010

FLORIAN DENNERT  and
RUDOLF GRÜBEL
FLORIAN DENNERT
Affiliation:
Institut für Mathematische Stochastik, Leibniz Universität Hannover, Postfach 6009, D-30060 Hannover, Germany (e-mail: [email protected], [email protected])
RUDOLF GRÜBEL
Affiliation:
Institut für Mathematische Stochastik, Leibniz Universität Hannover, Postfach 6009, D-30060 Hannover, Germany (e-mail: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

For random trees T generated by the binary search tree algorithm from uniformly distributed input we consider the subtree size profile, which maps k ∈ ℕ to the number of nodes in T that root a subtree of size k. Complementing earlier work by Devroye, by Feng, Mahmoud and Panholzer, and by Fuchs, we obtain results for the range of small k-values and the range of k-values proportional to the size n of T. In both cases emphasis is on the process view, i.e., the joint distributions for several k-values. We also show that the dynamics of the tree sequence lead to a qualitative difference between the asymptotic behaviour of the lower and the upper end of the profile.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 4 , July 2010 , pp. 561 - 578
Copyright
Copyright © Cambridge University Press 2010

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]Billingsley, P. (1968) Convergence of Probability Measures, Wiley, New York.Google Scholar
[2]Chauvin, B., Drmota, M. and Jabbour-Hattab, J. (2001) The profile of binary search trees. Ann. Appl. Probab. 11 1042–1062.CrossRefGoogle Scholar
[3]Dennert, F. (2009) Zufällige binäre Bäume: Algorithmen, Asymptotik und Statistik. Dissertation, Leibniz Universität Hannover.Google Scholar
[4]Devroye, L. (1991) Limit laws for local counters in random binary search trees. Random Struct. Alg. 2 303–315.CrossRefGoogle Scholar
[5]Drmota, M., Janson, S. and Neininger, R. (2008) A functional limit theorem for the profile of search trees. Ann. Appl. Probab. 18 288–333.Google Scholar
[6]Evans, S., Grübel, R. and Wakolbinger, A. (2009) Boundary theory for randomly growing binary trees. In preparation.Google Scholar
[7]Feng, Q., Mahmoud, H. M. and Panholzer, A. (2008) Phase changes in subtree varieties in random recursive and binary search trees. SIAM J. Discrete Math. 22 160–184.Google Scholar
[8]Fuchs, M. (2008) Subtree sizes in recursive trees and binary search trees: Berry–Esseen bounds and Poisson approximations. Combin. Probab. Comput. 17 661–680.CrossRefGoogle Scholar
[9]Fuchs, M., Hwang, H.-K. and Neininger, R. (2006) Profiles of random trees: Limit theorems for random recursive trees and binary search trees. Algorithmica 46 367–407.CrossRefGoogle Scholar
[10]Mahmoud, H. M. (1992) Evolution of Random Search Trees, Wiley, New York.Google Scholar
[11]Neininger, R. and Rüschendorf, L. (2004) A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14 378–418.CrossRefGoogle Scholar
[12]Sedgewick, R. and Flajolet, P. (1996) An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading.Google Scholar