  We consider two variational models for transport networks, an urban planning
and a branched transport model, in which the degree of network complexity and
ramification is governed by a small parameter $\varepsilon>0$. Smaller
$\varepsilon$ leads to finer ramification patterns, and we analyse how optimal
network patterns in a particular geometry behave as $\varepsilon\to0$ by
proving an energy scaling law. This entails providing constructions of
near-optimal networks as well as proving that no other construction can do
better.
  The motivation of this analysis is twofold. On the one hand, it provides a
better understanding of the transport network models; for instance, it reveals
qualitative differences in the ramification patterns of urban planning and
branched transport. On the other hand, several examples of variational pattern
analysis in the literature use an elegant technique based on relaxation and
convex duality. Transport networks provide a relatively simple setting to
explore variations and refinements of this technique, thereby increasing the
scope of its applicability.
