  In this paper we study the adjacency spectrum of families of finite rooted
trees with regular branching properties. In particular, we show that in the
case of constant branching, the eigenvalues are realized as the roots of a
family of generalized Fibonacci polynomials and produce a limiting distribution
for the eigenvalues as the tree depth goes to infinity. We indicate how these
results can be extended to periodic branching patterns and also provide a
generalization to higher order simplicial complexes.
