In probability theory, the Brownian tree, or Aldous tree, or Continuum Random Tree (CRT)[1] is a random real tree that can be defined from a Brownian excursion.
The Brownian tree was defined and studied by David Aldous in three articles published in 1991 and 1993.
This random tree has several equivalent definitions and constructions:[2] 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.
It is a fractal object which can be approximated with computers[3] or by physical processes with dendritic structures.
The following definitions are different characterisations of a Brownian tree, they are taken from Aldous's three articles.
[4][5][6] The notions of leaf, node, branch, root are the intuitive notions on a tree (for details, see real trees).
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
(which is to say the order of the nodes) and the edge lengths.
We define a probability law
on this space by:[clarification needed] where
depends not on the shape of the tree but rather on the total sum of all the edge lengths.
be a random metric space with the tree property, meaning there exists a unique path between two points of
points, chosen randomly under
In other words, the Brownian tree is defined from the laws of all the finite sub-trees one can generate from it.
with We then define an equivalence relation, noted
is then a distance on the quotient space
Definition — The random metric space
Consider a non-homogeneous Poisson point process N with intensity
is a Poisson variable with parameter
are exponential variables with decreasing means.
We then make the following construction: Definition — The closure
, equipped with the distance previously built, is called a Brownian tree.
This algorithm may be used to simulate numerically Brownian trees.
Consider a Galton-Watson tree whose reproduction law has finite non-zero variance, conditioned to have
be this tree, with the edge lengths divided by
In other words, each edge has length
The construction can be formalized by considering the Galton-Watson tree as a metric space or by using renormalized contour processes.
converges in distribution to a random real tree, which we call a Brownian tree.
Here, the limit used is the convergence in distribution of stochastic processes in the Skorokhod space (if we consider the contour processes) or the convergence in distribution defined from the Hausdorff distance (if we consider the metric spaces).