  We investigate the canonical Scheidegger Model of river network morphology
for the case of convergent and divergent underlying topography, by embedding it
on a cone. We find two distinct phases corresponding to few, long basins and
many, short basins, respectively, separated by a singularity in number of
basins, indicating a phase transition. Quantifying basin shape through Hack's
Law $l\sim a^h$ gives distinct values for the exponent $h$, providing a method
of testing our hypotheses. The generality of our model suggests implications
for vascular morphology, in particular differing number and shapes of arterial
and venous trees.
