The structure of Galton-Watson trees conditioned to be of a given size is well-understood. We provide yet another embedding theorem that permits us to obtain asymptotic probabilities of events that are determined by w...
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The structure of Galton-Watson trees conditioned to be of a given size is well-understood. We provide yet another embedding theorem that permits us to obtain asymptotic probabilities of events that are determined by what happens near the root of these trees. As an example, we derive the probability that a Galton-Watson tree is cut when each node is independently removed with probability p, where by cutting a tree we mean that every path from root to leaf must have at least one removed node.
The random split tree introduced by Devroye (1999) is considered. We derive a second order expansion for the mean of its internal path length and furthermore obtain a limit law by the contraction method. As an assumpt...
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The random split tree introduced by Devroye (1999) is considered. We derive a second order expansion for the mean of its internal path length and furthermore obtain a limit law by the contraction method. As an assumption we need the splitter having a Lebesgue density and mass in every neighborhood of 1. We use properly stopped homogeneous Markov chains, for which limit results in total variation distance as well as renewal theory are used. Furthermore, we extend this method to obtain the corresponding results for the Wiener index.
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