A tree-strip is the product of a finite set (graph) with an infinite
tree. For a tree-strip of finite cone type, the tree is of finite cone
type and constructed starting from a root with certain substitution rules.
For a vertex of such a tree one can consider the cone of descendants and
the term 'finite cone type' refers to the fact that there are only
finitely many different
non-isomorphic cones of descendants.
On a certain class of such trees we obtain absolutely continuous
spectrum for the Anderson model for low disorder. The proof is based in
an Implicit Function Theorem in a very abstract Banach space.

The most recent result considers the Fibonacci tree-strip which is quite
special.
For the original set up, an essential assumption needed is the fact that
each vertex has at least 2 children, i.e. the tree can not have short
line segments. This is the key assumption that excludes quasi-one
dimensional Anderson models on strips for which Anderson localization is
known.
The Fibonacci tree, whose number of vertices in the n-th generation
corresponds to the n-th Fibonacci number, violates this assumption. But
with certain modifications this special case can also be treated.
The Fibonacci tree-strip is the first tree-strip where the tree has
short line segments and absolutely continuous spectrum for random
operators could be established.