### Non-crossing trees revisited: cutting down and spanning subtrees

*Alois Panholzer*

#### Abstract

Here we consider two parameters for random non-crossing trees:

*(i)*the number of random cuts to destroy a size-n 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.Full Text: GZIP Compressed PostScript PostScript PDF original HTML abstract page