  A nearest-neighbor tight-binding model on a tree structure is investigated.
The full energy spectrum of the normalized Hamiltonian can be expressed in
terms of successively increasing number of contributions at any finite step of
construction of the tree, resulting in a causal chain. The degree of quantum
localization of any eigenstate, measured by the inverse participation ratio
(IPR), is also analytically expressed by means of terms in corresponding
eigenvalue chain. The resulting IPR scaling behavior is expressed by the tails
of eigenvalue chains as well.
