  A set of general allometric scaling laws is derived for different systems
represented by tree networks. The formulation postulates self-similar networks
with an arbitrary number of branches developed in each generation, and with an
inhomogeneous structure given by a fractal relation between successive
generations. Three idealized examples are considered: networks of masses,
electric resistors, and elastic springs, which obey a specific recurrence
relation between generations. The results can be generalized to networks made
with different elements obeying equivalent relations. The equivalent values of
the networks (total mass, resistance and elastic coefficient) are compared with
their corresponding spatial scales (length, cross-section and volume) in order
to derive allometric scaling laws. Under appropriate fractal-like
approximations of the length and cross-section of the branches, some allometric
exponents reported in the literature are recovered (for instance, the 3/4-law
of metabolism in biological organisms or the hydraulic conductivity scaling in
porous networks). The formulation allows different choices of the fractal
parameters, thus enabling the derivation of new power-laws not reported before.
