A comment on Horton's law of stream numbers

Horton's Law of Stream Numbers is shown to be internally inconsistent in the following sense: If there exists a large channel network of order S, with stream numbers that satisfy Horton's Law exactly, then the mean stream numbers for lower order networks contained in the large one will show definite deviations from Horton's Law. These mean stream numbers can be expressed by a recursive formula, which involves the probabilities Pij that a stream of order i terminates in a stream of order j > i. Reasonable assumptions about the nature of the Pij suggest that Horton curves (log Nk versus k) for the small order basins should be concave upwards, a result which is in general agreement with observation.