  The structure of a river network may be seen as a discrete set of nested
sub-networks built out of individual stream segments. These network components
are assigned an integral stream order via a hierarchical and discrete ordering
method. Exponential relationships, known as Horton's laws, between stream order
and ensemble-averaged quantities pertaining to network components are observed.
We extend these observations to incorporate fluctuations and all higher moments
by developing functional relationships between distributions. The relationships
determined are drawn from a combination of theoretical analysis, analysis of
real river networks including the Mississippi, Amazon and Nile, and numerical
simulations on a model of directed, random networks. Underlying distributions
of stream segment lengths are identified as exponential. Combinations of these
distributions form single-humped distributions with exponential tails, the sums
of which are in turn shown to give power law distributions of stream lengths.
Distributions of basin area and stream segment frequency are also addressed.
The calculations identify a single length-scale as a measure of size
fluctuations in network components. This article is the second in a series of
three addressing the geometry of river networks.
