Random laminations of the disk are the continuous limits of random non-crossing configurations of regular polygons. We provide an expository account on this subject. Initiated by the work of Aldous on the Brownian triangulation, this field now possesses many characters such as the random recursive triangulation, the stable laminations and the Markovian hyperbolic triangulation of the disk. We w...

We introduce a weighted model of random trees and analyze the asymptotic properties of their heights. Our framework encompasses most trees of logarithmic height that were introduced in the context of the analysis of algorithms or combinatorics. This allows us to state a sort of“master theorem”for the height of random trees, that covers binary search trees (Devroye, 1986), random recursive trees...

The hitting time Hxy between two vertices x and y of a graph is the average time that the standard simple random walk takes to get from x to y. In this paper, we study the distribution of the hitting time between two randomly chosen vertices of a random tree. We consider both uniformly random labelled trees and a more general model with vertex weights akin to simply generated trees. We show tha...

A widely used class of binary trees is studied in order to provide information useful in evaluating algorithms based on this storage structure. A closed form counting formula for the number of binary trees with n nodes and height k is developed and restated as a recursion more useful computationally . A generating function for the number of nodes given height is developed and used to find the a...

We propose an approach to analyze the asymptotic behavior of Pólya urns based on the contraction method. For this a combinatorial discrete time embedding of the evolution of the composition of the urn into random rooted trees is used. A decomposition of the trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each color. Ideas ...

If a recursive tree is selected uniformly at random from among all recursive trees on n vertices, then the distribution of the maximum in-degree is given asymptotically by the following theorem: for any 3xed integer d, Pn( 6 n + d)= exp(−2{ n}−d−1) + o(1) as n → ∞; where n = log2 n. (As usual, n denotes the greatest integer less than or equal to n, and { n}= n− n :) The proof makes extensive us...

We present the results concerning the embedding of trees into recursive circulants. Recursive circulant G(N,:,/) is a circulant graph with N vertices and jumps of powers of d. We present dilation 1 embedding of Fibonacci trees and full quaternaly trees in G(Z”, 2), and full binary trees and hinomial trees in G(2”,4).