Non-crossing trees revisited: cutting down and spanning subtrees

Here we consider two parameters for random non-crossing trees: i the number of random cuts to destroy a sizen non-crossing tree and 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 ii as a functional of a conditioned random walk, and although we do not have such an interpretation for parameter i , we obtain here limiting distributions, that are also arising as limits of some functionals of conditioned random walks.