Number of Complete N-ary Subtrees on Galton-Watson Family Trees |
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Authors: | George P Yanev Ljuben Mutafchiev |
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Institution: | (1) Department of Mathematics, University of South Florida, Tampa, FL 33620, USA;(2) American University in Bulgaria, 2700 Blagoevgrad, Bulgaria;(3) Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences, Sofia, Bulgaria |
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Abstract: | We associate with a Bienaymé-Galton-Watson branching process a family tree rooted at the ancestor. For a positive integer
, define a complete
-ary tree to be the family tree of a deterministic branching process with offspring generating function
. We study the random variables
and
counting the number of disjoint complete
-ary subtrees, rooted at the ancestor, and having height
and
, respectively. Dekking (1991) and Pakes and Dekking (1991) find recursive relations for
and
involving the offspring probability generation function (pgf) and its derivatives. We extend their results determining the
probability distributions of
and
. It turns out that they can be expressed in terms of the offspring pgf, its derivatives, and the above probabilities. We
show how the general results simplify in case of fractional linear, geometric, Poisson, and one-or-many offspring laws.
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Keywords: | Branching process Family tree Binary tree " target="_blank"> " target="_blank">gif" alt="$$N$$" align="middle" border="0"> -ary tree |
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