Trees, networks, and hydrology

This paper reviews theoretical and observational material on form and function of natural networks appeared in somewhat disparate contexts from physics to biology, whose study is related to hydrologic research. Moving from the exact result that drainage network configurations minimizing total energy dissipation are stationary solutions of the general equation describing landscape evolution, we discuss the properties and the dynamic origin of the scale-invariant structure of river patterns and its relation to optimal selection. We argue that at least in the fluvial landscape, nature works through imperfect searches for dynamically accessible optimal configurations and that purely random or deterministic constructs are clearly unsuitable to properly describe natural network forms. We also show that optimal networks are spanning loopless configurations only under precise physical requirements that arise under the constraints imposed by continuity. In the case of rivers, every spanning tree proves a local minimum of total energy dissipation. This is stated in a theorem form applicable to generic networks, suggesting that other branching structures occurring in nature (e.g., scale-free and looping) may possibly arise through optimality to different selective pressures. We thus conclude that one recurrent self-organized mechanism for the dynamic origin of fractal forms is the robust strive for imperfect optimality that we see embedded in many natural patterns, chief and foremost hydrologic ones.