This paper describes a probabilistic algorithm that, given a connected, undirected graph G with n vertices, produces a spanning tree of G chosen uniformly at random among the spanning trees of G. The expected running time is O(n logn) per generated tree for almost all graphs, and O(n3) for the worst graphs. Previously known deterministic algorithms and much more complicated and require O(n3) ti...

Random trees planning has been used to solve connectivity problems in a variety of problem domains. In particular, these approaches are some of the most successful approaches to robotic motion planning. In this work we examine the broad array of random-tree that have been proposed for motion planning. From this survey we distill a general algorithmic framework for random-tree planning that abst...

We design a compact adaptive tree structure for random access to coherent spatial data. Many papers explore compressed trees, but nearly all involve entropy coding of a sequential traversal, thus precluding random queries required by rendering algorithms. Instead, we use fixed-rate encoding for both the tree topology and its data. Key elements include the replacement of pointers by local offset...

We analyze the spectral distribution of the adjacency matrix and the graph Laplacian for a wide variety of random trees. Using soft arguments which seem to be applicable in a wide variety of settings, we show that the empirical spectral distribution for a number of random tree models, converges to a constant (model dependent) distribution. We also analyze the kernel of the spectrum and prove as...

We discuss several connections between discrete and continuous random trees. In the discrete setting, we focus on Galton-Watson trees under various conditionings. In particular, we present a simple approach to Aldous’ theorem giving the convergence in distribution of the contour process of conditioned Galton-Watson trees towards the normalized Brownian excursion. We also briefly discuss applica...

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...

Cartesian trees are binary search trees in which the nodes exhibit the heap property according to a second (priority) key. lithe search key and the priority key are independent, and the tree is built . based on n independent copies, Cartesian trees basically behave like ordinary random binary search trees . In this article, we analyze the expected behavior when the keys are dependent : in most ...

Preface These notes provide an elementary and self-contained introduction to branching random walks. Chapter 1 gives a brief overview of Galton–Watson trees, whereas Chapter 2 presents the classical law of large numbers for branching random walks. These two short chapters are not exactly indispensable, but they introduce the idea of using size-biased trees, thus giving motivations and an avant-...