1,612 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
Ubiquity of synonymity: almost all large binary trees are not uniquely identified by their spectra or their immanantal polynomials
There are several common ways to encode a tree as a matrix, such as the
adjacency matrix, the Laplacian matrix (that is, the infinitesimal generator of
the natural random walk), and the matrix of pairwise distances between leaves.
Such representations involve a specific labeling of the vertices or at least
the leaves, and so it is natural to attempt to identify trees by some feature
of the associated matrices that is invariant under relabeling. An obvious
candidate is the spectrum of eigenvalues (or, equivalently, the characteristic
polynomial). We show for any of these choices of matrix that the fraction of
binary trees with a unique spectrum goes to zero as the number of leaves goes
to infinity. We investigate the rate of convergence of the above fraction to
zero using numerical methods. For the adjacency and Laplacian matrices, we show
that that the {\em a priori} more informative immanantal polynomials have no
greater power to distinguish between trees
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