  We consider the number of nodes in the levels of unlabelled 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 unlabelled rooted trees and show that they
behave in this respect essentially like a conditioned Galton-Watson process.
