Rivers are open systems and achieve a steady state or grow allometrically, according to the general equation y = axb. This equation yields a straight line on double logarithmic paper and reflects a lognormal or Yule stochastic process. Horton's laws are shown to be statements of these processes, and successive stream orders are seen to be new logarithmic cycles to the base of the bifurcation ratio for a river system. A new absolute order is suggested, derived by raising the bifurcation ratio to successive integer powers equivalent to stream order minus one.

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