Hostname: page-component-6766d58669-r8qmj Total loading time: 0 Render date: 2026-05-23T03:47:46.048Z Has data issue: false hasContentIssue false
  • English
  • Français

Profiles of random trees: correlation and width of random recursive trees and binary search trees

Part of: Linear inference, regression Graph theory Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

Michael Drmota  and
Hsien-Kuei Hwang
Michael Drmota*
Affiliation:
Technische Universität Wien
Hsien-Kuei Hwang*
Affiliation:
Academia Sinica
*
∗ Postal address: Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/118, 1040 Wien, Austria.
∗∗ Postal address: Institute of Statistical Science, Academia Sinica, Taipei, 115, Taiwan. Email address: hkhwang@stat.sinica.edu.tw
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In a tree, a level consists of all those nodes that are the same distance from the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees. These coefficients undergo sharp sign-changes when one level is fixed and the other is varying. We also propose a new means of deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by singularity analysis.

Keywords

Correlation coefficientrandom recursive treerandom binary search treewidthtotal path lengthsign-change

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 62J10: Analysis of variance and covariance 05C05: Trees

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 2 , June 2005 , pp. 321 - 341
Copyright
Copyright © Applied Probability Trust 2005 

References

Arya, S., Golin, M. J. and Mehlhorn, K. (1999). On the expected depth of random circuits. Combinatorics Prob. Comput. 8, 209–228.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Booth, A. D. and Colin, J. T. (1960). On the efficiency of a new method of dictionary construction. Inf. Control 3, 327–334.Google Scholar
Brown, G. G. and Shubert, B. O. (1984). On random binary trees. Math. Operat. Res. 9, 43–65.Google Scholar
Chauvin, B., Drmota, M. and Jabbour-Hattab, J. (2001). The profile of binary search trees. Ann. Appl. Prob. 11, 1042–1062.Google Scholar
Dobrow, R. P. and Fill, J. A. (1999). Total path length for random recursive trees. Combinatorics Prob. Comput. 8, 317–333.Google Scholar
Drmota, M. and Hwang, H.-K. (2005). Bimodality and phase transitions in the profile variance of random binary search trees. To appear in SIAM J. Discrete Math.Google Scholar
Flajolet, P. and Odlyzko, A. M. (1990). Singularity analysis of generating functions. SIAM J. Discrete Math. 3, 216–240.Google Scholar
Françon, J. (1977). On the analysis of algorithms for trees. Theoret. Comput. Sci. 4, 155–169.Google Scholar
Fuchs, M., Hwang, H.-K. and Neininger, R. (2005). Profiles of random trees: limit theorems for random recursive trees and binary search trees. Submitted. Available at http://algo.stat.sinica.edu.tw/.Google Scholar
Gastwirth, J. L. (1977). A probability model of a pyramid scheme. Amer. Statistician 31, 79–82.Google Scholar
Grossman, R. and Larson, R. G. (1989). Hopf-algebraic structure of families of trees. J. Algebra 126, 184–210.Google Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Probability and Mathematical Statistics. Academic Press, New York.Google Scholar
Hibbard, T. N. (1962). Some combinatorial properties of certain trees with applications to searching and sorting. J. Assoc. Comput. Mach. 9, 13–28.Google Scholar
Hwang, H.-K. (1995). Asymptotic expansions for the Stirling numbers of the first kind. J. Combinatorial Theory A 71, 343–351.Google Scholar
Hwang, H.-K. (1997). Asymptotic estimates for elementary probability distributions. Studies Appl. Math. 99, 393–417.Google Scholar
Knuth, D. E. (1997). The Art of Computer Programming, Vol. 3, Sorting and Searching, 2nd edn. Addison-Wesley, Reading, MA.Google Scholar
Lynch, W. C. (1965). More combinatorial properties of certain trees. Comput. J. 7, 299–302.Google Scholar
Mahmoud, H. M. (1991). Limiting distributions for path lengths in recursive trees. Prob. Eng. Inf. Sci. 5, 53–59.Google Scholar
Mahmoud, H. M. (1992). Evolution of Random Search Trees. John Wiley, New York.Google Scholar
Meir, A. and Moon, J. W. (1974). Cutting down recursive trees. Math. Biosci. 21, 173–181.Google Scholar
Mitchell, S. L., Cockayne, E. J. and Hedetniemi, S. T. (1979). Linear algorithms on recursive representations of trees. J. Comput. System Sci. 18, 76–85.Google Scholar
Moon, J. W. (1974). The distance between nodes in recursive trees. In Combinatorics (London Math. Soc. Lecture Notes Ser. 13), eds McDonough, T. P. and Marron, V. C., Cambridge University Press, pp. 125–132.Google Scholar
Na, H. S. and Rapoport, A. (1970). Distribution of nodes of a tree by degree. Math. Biosci. 6, 313–329.Google Scholar
Tapia, M. A. and Myers, B. R. (1967). Generation of concave node-weighted trees. IEEE Trans. Circuits Systems 14, 229–230.Google Scholar
Van der Hofstad, R., Hooghiemstra, G. and van Mieghem, P. (2002). On the covariance of the level sizes in random recursive trees. Random Structures Algorithms 20, 519–539.Google Scholar
Windley, P. F. (1960). Trees, forests and rearranging. Comput. J. 3, 84–88.Google Scholar