Open Access
August, 1991 On the Distribution of Leaves in Rooted Subtrees of Recursive Trees
Hosam M. Mahmoud, R. T. Smythe
Ann. Appl. Probab. 1(3): 406-418 (August, 1991). DOI: 10.1214/aoap/1177005874

Abstract

We study the structure of Tn(k), the subtree rooted at k in a random recursive tree of order n, under the assumption that k is fixed and n. Employing generalized Polya urn models, exact and limiting distributions are derived for the size, the number of leaves and the number of internal nodes of Tn(k). The exact distributions are given by intricate formulas involving Eulerian numbers, but a recursive argument based on the urn model suffices for establishing the first two moments of the above-mentioned random variables. Known results show that the limiting distribution of the size of Tn(k), normalized by dividing by n is Beta(1,k1). A martingale central limit argument is used to show that the difference between the number of leaves and the number of internal nodes of Tn(k), suitably normalized, converges to a mixture of normals with a Beta(1,k1) as the mixing density. The last result allows an easy determination of limiting distributions of suitably normalized versions of the number of leaves and the number of internal nodes of Tn(k).

Citation

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Hosam M. Mahmoud. R. T. Smythe. "On the Distribution of Leaves in Rooted Subtrees of Recursive Trees." Ann. Appl. Probab. 1 (3) 406 - 418, August, 1991. https://doi.org/10.1214/aoap/1177005874

Information

Published: August, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0738.05034
MathSciNet: MR1111525
Digital Object Identifier: 10.1214/aoap/1177005874

Subjects:
Primary: 05C05
Secondary: 60G42 , 68E05

Keywords: generalized Polya urn models , martingale central limit theorem , recursive trees , rooted subtrees

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.1 • No. 3 • August, 1991
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