However, in river systems there is a great deal of evidence for power-law and multifractal behaviour rather than exponential.
There is a whole book that describes a variety of aspects of this (Rodriguez-Iturbe and Rinaldo, 1997).
In particular, perhaps the network width function is important in this context. It gives the number of links in the basin as a function of distance from the outlet and has been shown to be multifractal. The width function has been used for flood prediction and for defining the geomorphic instantaneous unit hydrograph.
I., Rinaldo I. 1997. Fractal River Basins: Chance and self-organization.
Cambridge University Press.