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Exact formulas for the variance of several balance indices under the Yule model

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Abstract

One of the main applications of balance indices is in tests of null models of evolutionary processes. The knowledge of an exact formula for a statistic of a balance index, holding for any number \(n\) of leaves, is necessary in order to use this statistic in tests of this kind involving trees of any size. In this paper we obtain exact formulas for the variance under the Yule model of the Sackin, the Colless and the total cophenetic indices of binary rooted phylogenetic trees with \(n\) leaves.

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Acknowledgments

This research has been partially supported by the Spanish government and the UE FEDER program, through projects MTM2009-07165 and TIN2011-15874-E. We thank J. Miró and M. Lewis for several comments on a previous version of this manuscript. We also thank the comments and suggestions of the associate editor and the reviewers, that have led to a substantial improvement of this paper. Many computations in this paper have been carried out or checked with the aid of Mathematica.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of the Balearic Islands, 07122 , Palma, Spain

    Gabriel Cardona, Arnau Mir & Francesc Rosselló

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  1. Gabriel Cardona
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  2. Arnau Mir
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  3. Francesc Rosselló
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Correspondence to Francesc Rosselló.

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Cardona, G., Mir, A. & Rosselló, F. Exact formulas for the variance of several balance indices under the Yule model. J. Math. Biol. 67, 1833–1846 (2013). https://doi.org/10.1007/s00285-012-0615-9

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  • DOI: https://doi.org/10.1007/s00285-012-0615-9

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