  In the 1980s an important goal of the emergent field of fractals was to
determine the relationships between their physical and geometrical properties.
The fractal-Einstein and Alexander-Orbach laws, which interrelate electrical,
diffusive and fractal properties, are two key theories of this type. Here we
settle a long standing controversy about their exactness by showing that the
properties of a class of fractal trees violate both laws. A new formula is
derived which unifies the two classical results by proving that if one holds,
then so must the other, and resolves a puzzling discrepancy in the properties
of Eden trees and diffusion limited aggregates. The failure of the classical
laws is attributed to anisotropic exploration of the network by a random
walker. The occurrence of this newly revealed behaviour means that numerous
theories, such as recent first passage time results, are restricted to a
narrower range of networks than previously thought.
