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Limiting Distributions for Path Lengths in Recursive Trees

Published online by Cambridge University Press:  27 July 2009

Hosam M. Mahmoud
Hosam M. Mahmoud
Affiliation:
Department of Statistics/Computer & In formation Systems George Washington University Washington, D.C. 20052, U.S.A.
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Abstract

The depth of insertion and the internal path length of recursive trees are studied. Luc Devroye has recently shown that the depth of insertion in recursive trees is asymptotically normal. We give a direct alternative elementary proof of this fact. Furthermore, via the theory of martingales, we show that In, the internal path length of a recursive tree of order n, converges to a limiting distribution. In fact, we show that there exists a random variable I such that (In – n In n)/n→I almost surely and in quadratic mean, as n → α. The method admits, in passing, the calculation of the first two moments of In.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 5 , Issue 1 , January 1991 , pp. 53 - 59
Copyright
Copyright © Cambridge University Press 1991

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