Tree Densities

Why

As with distributions, if a density factors according to a tree rooted at a particular vertex, it factors according to a tree rooted at any vertex. So, as before, the property of factoring according to a rooted tree is really a property of factoring according to a tree.

Defining result

If a density factors according to a tree rooted at a vertex it factors according to that tree rooted at any vertex.

Undirected definition

A density factors according to a tree if it factors according to the tree rooted at any vertex.

Existences and uniqueness

Trees are not a property of densities, since there is no one-to-one correspondence, as demonstrated by the following propositions.

Existence

A density $f$ need not factor according to a tree.

Uniqueness

A density $f$ may factor according to multiple trees.