It is proved that the moments of the width of Galton-Watson trees of size n and with ospring variance are asymptotically given by ( n) mp where mp are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a tightness estimate. The method is quite general and we state some further applications. 1.

Abstract. We consider the number of nodes in the levels of unlabeled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned Galton-Watson trees and forests to the case of unlabeled rooted trees and show that they behave in this respect essentially like a conditioned Galton-Watson process. 1.

We consider the number of nodes in the levels of unlabeled rooted random trees and show that the joint distribution of several level sizes (where the level number is scaled by √ n) weakly converges to the distribution of the local time of a Brownian excursion evaluated at the times corresponding to the level numbers. This extends existing results for

Abstract. We investigate the profile of random Pólya trees of size n when only nodes of degree d are counted in each level. It is shown that, as in the case where all nodes contribute to the profile, the suitably normalized profile process converges weakly to a Brownian excursion local time. Moreover, we investigate the joint distribution of the number of nodes of degree d1 and d2 in the levels of the tree. 1.