Geodesic
Non-crossing trees revisited: cutting down and spanning subtrees
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03), DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03) (2003).

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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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     author = {Panholzer, Alois},
     title = {Non-crossing trees revisited: cutting down and spanning subtrees},
     journal = {Discrete mathematics & theoretical computer science},
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     volume = {DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03)},
     year = {2003},
     doi = {10.46298/dmtcs.3327},
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     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3327/}
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Panholzer, Alois. Non-crossing trees revisited: cutting down and spanning subtrees. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03), DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03) (2003). doi : 10.46298/dmtcs.3327. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3327/

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