  We prove that certain random models associated with radial, tree-like, rooted
quantum graphs exhibit Anderson localization at all energies. The two main
examples are the random length model (RLM) and the random Kirchhoff model
(RKM). In the RLM, the lengths of each generation of edges form a family of
independent, identically distributed random variables (iid). For the RKM, the
iid random variables are associated with each generation of vertices and
moderate the current flow through the vertex. We consider extensions to various
families of decorated graphs and prove stability of localization with respect
to decoration. In particular, we prove Anderson localization for the random
necklace model.
