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On the Number of t-Ary Trees with a Given Path Length

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Abstract

We show that the number of t-ary trees with path length equal to p is

$\exp({{(\alpha {p}/{\log p})}(1+o(1))}),$

where \(\alpha=h(t^{-1})t\log t\) and \(h(x)={-}x\log x {-}(1{-}x)\log (1{-}x).\) Besides its intrinsic combinatorial interest, the question recently arose in the context of information theory, where the number of t-ary trees with path length p estimates the number of universal types, or, equivalently, the number of different possible Lempel-Ziv '78 dictionaries for sequences of length p over an alphabet of size t.

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Authors and Affiliations

  1. Mathematical Sciences Research Institute, Berkeley, CA 94720, USA

    Gadiel Seroussi

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  1. Gadiel Seroussi
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Correspondence to Gadiel Seroussi.

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Seroussi, G. On the Number of t-Ary Trees with a Given Path Length. Algorithmica 46, 557–565 (2006). https://doi.org/10.1007/s00453-006-0122-8

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  • DOI: https://doi.org/10.1007/s00453-006-0122-8

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