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Limit laws for the Randić index of random binary tree models

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Abstract

We investigate the Randić index of random binary trees under two standard probability models: the one induced by random permutations and the Catalan (uniform). In both cases the mean and variance are computed by recurrence methods and shown to be asymptotically linear in the size of the tree. The recursive nature of binary search trees lends itself in a natural way to application of the contraction method, by which a limit distribution (for a suitably normalized version of the index) is shown to be Gaussian. The Randić index (suitably normalized) is also shown to be normally distributed in binary Catalan trees, but the methodology we use for this derivation is singularity analysis of formal generating functions.

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Authors and Affiliations

  1. Department of Statistics and Finance, University of Science and Technology of China, Hefei, 230026, China

    Qunqiang Feng

  2. Department of Statistics, The George Washington University, Washington, DC, 20052, USA

    Hosam M. Mahmoud

  3. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstr., 8-10/104, 1040, Wien, Austria

    Alois Panholzer

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  1. Qunqiang Feng
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  2. Hosam M. Mahmoud
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  3. Alois Panholzer
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Correspondence to Hosam M. Mahmoud.

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Feng, Q., Mahmoud, H.M. & Panholzer, A. Limit laws for the Randić index of random binary tree models. AISM 60, 319–343 (2008). https://doi.org/10.1007/s10463-006-0107-z

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  • DOI: https://doi.org/10.1007/s10463-006-0107-z

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