Distances between pairs of vertices and vertical profile in conditioned Galton--Watson trees

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Devroye, Luc
Janson, Svante
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Abstract
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We consider a conditioned Galton-Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis. Moreover, the second proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet-Melou and Janson saying that the vertical profile of a randomly labelled conditioned Galton-Watson tree converges in distribution, after suitable normalization, to the density of ISE (Integrated Superbrownian Excursion).
Comment: 16 pages
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Mathematics - Probability, Mathematics - Combinatorics, 60C05, 05C05
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