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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 $T^{(k)}_n$, the subtree rooted at $k$ in a random recursive tree of order $n$, under the assumption that $k$ is fixed and $n \rightarrow \infty$. 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 $T^{(k)}_n$. 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 $T^{(k)}_n$, normalized by dividing by $n$ is $\operatorname{Beta}(1, k - 1)$. A martingale central limit argument is used to show that the difference between the number of leaves and the number of internal nodes of $T^{(k)}_n$, suitably normalized, converges to a mixture of normals with a $\operatorname{Beta}(1, k - 1)$ 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 $T^{(k)}_n$.

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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

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Vol.1 • No. 3 • August, 1991
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