Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T17:44:49.819Z Has data issue: false hasContentIssue false
  • English
  • Français

On the distribution of distances in recursive trees

Part of: Graph theory Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Robert P. Dobrow
Robert P. Dobrow*
Affiliation:
National Institute of Standards and Technology
*
∗Postal address: Division of Mathematics and Computer Science, Northeast Missouri State University, Kirksville, MO 63501, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree Tn with n labeled nodes is a recursive tree if n = 1, or n > 1 and Tn can be constructed by joining node n to a node of some recursive tree Tn–1. For arbitrary nodes i < n in a random recursive tree we give the exact distribution of Xi,n, the distance between nodes i and n. We characterize this distribution as the convolution of the law of Xi,j+1 and n – i – 1 Bernoulli distributions. We further characterize the law of Xi,j+1 as a mixture of sums of Bernoullis. For i = in growing as a function of n, we show that is asymptotically normal in several settings.

Keywords

RECURSIVE TREESSTIRLING NUMBERS OF THE FIRST KIND

MSC classification

Primary: 05C05: Trees
Secondary: 60C05: Combinatorial probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996 , pp. 749 - 757
Copyright
Copyright © Applied Probability Trust 1996 

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

Devroye, L. (1988) Applications of the theory of records in the study of random trees. Acta Info. 26, 123–130.Google Scholar
Gastwirth, J. L. and Bhattacharya, P. K. (1984) Two probability models of pyramid or chain letter schemes demonstrating that their promotional claims are unreliable. Operat. Res. 32, 527–536.Google Scholar
Goldie, C. M. (1989) Records, permutations and greatest convex minorants. Math. Proc. Camb. Phil. Soc. 106, 169–177.CrossRefGoogle Scholar
Graham, R. L., Knuth, D. E. and Patashnik, O. (1989) Concrete Mathematics. Addison-Wesley, Reading, MA.Google Scholar
Mahmoud, H. M. (1991) Limiting distributions for path lengths in recursive trees. Prob. Eng. Info. Sci. 5, 53–59.Google Scholar
Mahmoud, H. M. and Smythe, R. T. (1992) Asymptotic joint normality of outdegrees of nodes in random recursive trees. Rand. Struct. Alg. 3, 255–266.Google Scholar
Moon, J. W. (1974) The distance between nodes in recursive trees. London Math. Soc. Lecture Notes 13, 125–132.Google Scholar
Najock, D. and Heyde, C. C. (1982) On the number of terminal vertices in certain random trees with an application to stemma construction in philology. J. Appl. Prob. 19, 675–680.Google Scholar
Szymanski, J. (1990) On the maximum degree and the height of a random recursive tree. In Random Graphs '87. ed. Karonski, M. and Rucinski, A. pp. 313–324.Google Scholar