  The eigenvalues of the normalized Laplacian matrix of a network plays an
important role in its structural and dynamical aspects associated with the
network. In this paper, we study the spectra and their applications of
normalized Laplacian matrices of a family of fractal trees and dendrimers
modeled by Cayley trees, both of which are built in an iterative way. For the
fractal trees, we apply the spectral decimation approach to determine
analytically all the eigenvalues and their corresponding multiplicities, with
the eigenvalues provided by a recursive relation governing the eigenvalues of
networks at two successive generations. For Cayley trees, we show that all
their eigenvalues can be obtained by computing the roots of several
small-degree polynomials defined recursively. By using the relation between
normalized Laplacian spectra and eigentime identity, we derive the explicit
solution to the eigentime identity for random walks on the two treelike
networks, the leading scalings of which follow quite different behaviors. In
addition, we corroborate the obtained eigenvalues and their degeneracies
through the link between them and the number of spanning trees.
