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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.
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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 

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