probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, the Brownian tree, or Aldous tree, or Continuum Random Tree (CRT) is a special case from random real trees which may be defined from a

Brownian excursion In probability theory a Brownian excursion process is a stochastic process that is closely related to a Wiener process (or Brownian motion). Realisations of Brownian excursion processes are essentially just realizations of a Wiener process select ...

. The Brownian tree was defined and studied by David Aldous in three articles published in 1991 and 1993. This tree has since then been generalized. This random tree has several equivalent definitions and constructions: using sub-trees generated by finitely many leaves, using a Brownian excursion, Poisson separating a straight line or as a limit of Galton-Watson trees. Intuitively, the Brownian tree is a binary tree whose nodes (or branching points) are

dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...

in the tree; which is to say that for any distinct two points of the tree, there will always exist a node between them. It is a

fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...

object which can be approximated with computers or by physical processes with dendritic structures.

Definitions

The following definitions are different characterisations of a Brownian tree, they are taken from Aldous's three articles. The notions of ''leaf, node, branch, root'' are the intuitive notions on a tree (for details, see real trees).

Finite-dimensional laws

This definition gives the finite-dimensional laws of the subtrees generated by finitely many leaves. Let us consider the space of all binary trees with

k

leaves numbered from

1

k

. These trees have

2k-1

edges with lengths

(\ell_1,\dots,\ell_)\in \R_+^

. A tree is then defined by its shape

\tau

(which is to say the order of the nodes) and the edge lengths. We define a probability law

\mathbb

of a random variable

(T,(L_i)_)

on this space by: :

\mathbb P(T=\tau \,, \, L_i\in