We survey recent developments about random real trees, whose prototype is the Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly explain the formalism of real trees, which yields a neat presentation of the theory and in particular of the relations between discrete Galton-Watson trees and continuous random trees. We then discuss the particular class of self-similar random real ...

In arXiv:1609.05666v1 [math.PR] a functional limit theorem was proved. It states that symmetric processes associated with resistance metric measure spaces converge when the underlying respect to Gromov-Hausdorff-vague topology, and certain uniform recurrence condition is satisfied. Such finds particularly nice applications if space tree. To illustrate this, we state theorems in old new examples...

Limit theorems are established for some functionals of the distances between two nodes in weighted random b-ary recursive trees. We consider the depth of the nth node and of a random node, the distance between two random nodes, the internal path length and the Wiener index. As application these limit results imply by an imbedding argument corresponding limit theorems for further classes of rand...

The Random Projection Tree (RPTREE) structures proposed in [1] are space partitioning data structures that automatically adapt to various notions of intrinsic dimensionality of data. We prove new results for both the RPTREE-MAX and the RPTREE-MEAN data structures. Our result for RPTREE-MAX gives a nearoptimal bound on the number of levels required by this data structure to reduce the size of it...

A random suffix search tree is a binary search tree constructed for the suffixes Xi = 0.BiBi+1Bi+2 . . . of a sequence B1, B2, B3., . . . of independent identically distributed random b-ary digits Bj . Let Dn denote the depth of the node for Xn in this tree when B1 is uniform on Zb. We show that for any value of b > 1, EDn = 2 log n + O(log log n), just as for the random binary search tree. We ...

We study fragmentation of a random recursive tree into a forest by repeated removal of nodes. The initial tree consists of N nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly-selected existing node. As nodes are removed from the tree, one at a time, the tree dissolves into an ensemble of separate trees, namely, a forest. We study statistical pr...

2 Binary search trees 2 2.1 Definition of a binary search tree . . . . . . . . . . . . . . . . . . 2 2.2 Profile of a binary search tree . . . . . . . . . . . . . . . . . . . . 3 2.2.1 Level polynomial. BST martingale . . . . . . . . . . . . . 3 2.2.2 Embedding in continuous time. Yule tree . . . . . . . . . . 6 2.2.3 Connection Yule tree binary search tree . . . . . . . . . . 7 2.2.4 Asymptoti...

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

We consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We...