685 research outputs found
The Shape of Unlabeled Rooted Random Trees
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.Comment: 34 pages, 1 figur
Graph limits of random graphs from a subset of connected -trees
For any set of non-negative integers such that and , we consider a random --tree that is uniformly selected from all connected -trees of
vertices where the number of -cliques that contain any fixed -clique
belongs to . We prove that , scaled by
where is the -th Harmonic
number and , converges to the Continuum Random Tree
. Furthermore, we prove the local convergence of the
rooted random --tree to an infinite but
locally finite random --tree .Comment: 21 pages, 6 figure
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