Allometric scaling laws derived from symmetric tree networks

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.