In
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
leaves numbered from
to
. These trees have
edges with lengths
. A tree is then defined by its shape
(which is to say the order of the nodes) and the edge lengths. We define a
probability law of a random variable
on this space by:
:
where
.
In other words,
depends not on the shape of the tree but rather on the total sum of all the edge lengths.
In other words, the Brownian tree is defined from the laws of all the finite sub-trees one can generate from it.
Continuous tree
The Brownian tree is a
real tree 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 ...
(see characterisation 4 in
Real tree).
Let
be a Brownian excursion. Define a
pseudometric on