Continuum diffusion on networks: trees with hyperbranched trunks and fractal branches.

The probability that a random walker returns to its origin for large times scales as t;{-d[over ]2} , where d[over ] is the spectral dimension. We calculate d[over ] for a class of tree structures using a renormalization technique on an infinite continued fraction. We consider a wide range of homogeneous networks based on replacing the branches of a self-similar tree with arbitrary fractals and composite fractals. We also consider a new class of inhomogeneous hyperbranched trees.