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Non-crossing trees revisited: cutting down and spanning subtrees

Abstract : Here we consider two parameters for random non-crossing trees: $\textit{(i)}$ the number of random cuts to destroy a size-$n$ non-crossing tree and $\textit{(ii)}$ the spanning subtree-size of $p$ randomly chosen nodes in a size-$n$ non-crossing tree. For both quantities, we are able to characterise for $n → ∞$ the limiting distributions. Non-crossing trees are almost conditioned Galton-Watson trees, and it has been already shown, that the contour and other usually associated discrete excursions converge, suitable normalised, to the Brownian excursion. We can interpret parameter $\textit{(ii)}$ as a functional of a conditioned random walk, and although we do not have such an interpretation for parameter $\textit{(i)}$, we obtain here limiting distributions, that are also arising as limits of some functionals of conditioned random walks.
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https://hal.inria.fr/hal-01183922
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Alois Panholzer. Non-crossing trees revisited: cutting down and spanning subtrees. Discrete Random Walks, DRW'03, 2003, Paris, France. pp.265-276, ⟨10.46298/dmtcs.3327⟩. ⟨hal-01183922⟩

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