Inferring scaling laws of branching networks from their geometry by global optimization

Optimality principles are widely used to understand branching networks, yet they are tested on individual branchings that cannot reliably distinguish competing scaling laws. We formulate the missing link between the laws and measurements by mapping each law to its predicted branch points through global optimization. Vascular branch points closely match weighted-length optima when each vessel is allowed its own fitted weight. Inferred scaling laws reveal type-specific geometric signatures hidden from local analysis. They beat all tested laws, including Murray's law in 1,679 of 1,744 held-out hepatic and pulmonary trees. For coronary arteries on the heart surface, an exact discrete optimization model extends the framework beyond convex geometry and supports area-preserving scaling. The framework also identifies distinct laws for airways and plant shoots while rejecting weighted-length optimality for neuronal axons. Global optimization turns network geometry into a quantitative test of the principles that govern branching networks.