The well-known Horton's laws are empirical observations on how the means of measurements for river networks and basins vary with Horton-Strahler order. It is now known that these laws are a consequence of an average-sense self-similarity in the bifurcation structure of river networks. In this paper we present a reformulation of Horton's laws which generalizes the familiar scaling of first moments, or means, to scaling of entire distributions. We also present extensive data analysis which supports this reformulation and show that this feature is also exhibited by Shreve's well-known random topology model. ¿ 1999 American Geophysical Union