Tree Distributions
Why
It happens that if a distribution factors according to a tree rooted at a particular vertex, it factors according to a tree rooted at any vertex. So the property of factoring according to a rooted tree is really property of factoring according to a tree.
Definition
Defining Result
Let $A_1, \dots , A_n$ be finite non-empty sets
and define $A = \prod_{i = 1}^{n} A_n$.
Let $p: A \to [0, 1]$ be a distribution and
let $T$ be a tree on $\set{1, \dots , n}$.
If a distribution factors according to a tree
rooted at a vertex it factors according to that
tree rooted at any vertex.
Undirected definition
A distribution $p$ factors according to the tree $T$ if it factors according to the $T$ rooted at any vertex.
Existence and uniqueness
Trees are not a property of distributions, since there is no one-to-one correspondence, as demonstrated by the following propositions.
Existence
A distribution $p$ need not factor according to a tree.
Uniqueness
A distribution $p$ may factor according to multiple trees.