Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T22:50:07.900Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Properties of Protected Nodes in Random Recursive Trees

Part of: Limit theorems Combinatorial probability

Published online by Cambridge University Press:  30 January 2018

Hosam M. Mahmoud  and
Mark D. Ward
Hosam M. Mahmoud*
Affiliation:
The George Washington University
Mark D. Ward*
Affiliation:
Purdue University
*
∗ Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA. Email address: hosam@gwu.edu
∗∗ Postal address: Department of Statistics, Purdue University, West Lafayette, IN 47907-2067, USA. Email address: mdw@purdue.edu
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.

We investigate protected nodes in random recursive trees. The exact mean of the number of such nodes is obtained by recurrence, and a linear asymptotic equivalent follows. A nonlinear recurrence for the variance shows that the variance grows linearly, too. It follows that the number of protected nodes in a random recursive tree, upon proper scaling, converges in probability to a constant.

Keywords

Recursive treerandom structurecombinatorial probability

MSC classification

Primary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 290 - 297
Copyright
© Applied Probability Trust 

References

Bergeron, F., Flajolet, P. and Salvy, B. (1992). “Varieties of increasing trees.” In (CAAP '92) Lecture Notes Comput. Sci. 581, Springer, Berlin, pp. 24–48.Google Scholar
Cheon, G.-S. and Shapiro, L. W. (2008). “Protected points in ordered trees.” Appl. Math. Lett. 21, 516–520.Google Scholar
Du, R. R. X. and Prodinger, H. (2012). “Notes on protected nodes in digital search trees.” Appl. Math. Lett. 25, 1025–1028.CrossRefGoogle Scholar
Flajolet, P. and Sedgewick, R. (2009). “Analytic Combinatorics.” Cambridge University Press.Google Scholar
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
Mansour, T. (2011). “Protected points in k-ary trees.” Appl. Math. Lett. 24, 478–480.Google Scholar
Smythe, R. T. and Mahmoud, H. M. (1995). “A survey of recursive trees.” Theory Prob. Math. Statist. 51, 1–27.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