Biological transport networks — blood vessels, leaf veins, river deltas — overwhelmingly use binary branching. Each junction splits a single channel into exactly two. Not three. Not four. Two. The universality of binary branching has been noted but not explained from first principles.
Wang, Ristroph, and Alben derive it variationally. They construct a game-theoretic framework where the network must minimize a cost function (material cost of the channels plus dissipation cost of the transport) subject to the constraint of connecting a source to distributed sinks. The optimal branching topology is the Nash equilibrium of this game.
The result: binary branching is the unique equilibrium. Higher-order branching (three or more children per junction) is always suboptimal. The proof doesn't depend on specific cost functions — it holds for any cost that combines material and transport penalties with reasonable monotonicity assumptions.
The mechanism is geometric. At a ternary junction, one of the three child branches always has an incentive to migrate toward one of the other two, forming a binary cascade that reduces total path length. The ternary point is unstable; any perturbation resolves it into two binary junctions. The instability is topological, not quantitative — it doesn't depend on the magnitudes of the costs, only on their structure.
This explains why Murray's law and its generalizations all produce binary networks: they're specific cost functions within a class that universally yields binary branching at equilibrium. The binary topology is not a consequence of the particular physics (viscous flow, elastic deformation, etc.) but of the variational structure itself.
Binary branching is not what happens to work. It's what must happen. The structure is forced by the game.