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Profile of Random Exponential Recursive Trees

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  • Hosam Mahmoud

    (The George Washington University)

Abstract

We introduce the random exponential recursive tree in which at each point of discrete time every node recruits a child (new leaf) with probability p, or fails to do so with probability 1 − p. We study the distribution of the size of these trees and the average level composition, often called the profile. We also study the size and profile of an exponential version of the plane-oriented recursive tree (PORT), wherein every insertion positions in the “gaps” between the edges recruits a child (new leaf) with probability p, or fails to do so with probability 1 − p. We use martingales in conjunction with distributional equations to establish strong laws for the size of both exponential flavors; in both cases, the limit laws are characterized by their moments. Via generating functions, we get an exact expression for the average expectation of the number of nodes at each level. Asymptotic analysis reveals that the most populous level is p p + 1 n $\frac {p}{p+1} n$ in exponential recursive trees, and is p 2 p + 1 n $\frac {p}{2p+1} n$ in exponential PORTs.

Suggested Citation

  • Hosam Mahmoud, 2022. "Profile of Random Exponential Recursive Trees," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 259-275, March.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:1:d:10.1007_s11009-020-09831-9
    DOI: 10.1007/s11009-020-09831-9
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    References listed on IDEAS

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    1. Panpan Zhang & Hosam M. Mahmoud, 2020. "On Nodes of Small Degrees and Degree Profile in Preferential Dynamic Attachment Circuits," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 625-645, June.
    2. Yarong Feng & Hosam Mahmoud, 2018. "Profile of Random Exponential Binary Trees," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 575-587, June.
    3. Hosam M. Mahmoud, 2010. "The Power of Choice in the Construction of Recursive Trees," Methodology and Computing in Applied Probability, Springer, vol. 12(4), pages 763-773, December.
    4. R. M. D'Souza & P. L. Krapivsky & C. Moore, 2007. "The power of choice in growing trees," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 59(4), pages 535-543, October.
    5. J. L. Gastwirth & P. K. Bhattacharya, 1984. "Two Probability Models of Pyramid or Chain Letter Schemes Demonstrating that Their Promotional Claims are Unreliable," Operations Research, INFORMS, vol. 32(3), pages 527-536, June.
    Full references (including those not matched with items on IDEAS)

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