We present a scaling argument based on network volume minimization that explains how material-distributing and -collecting branching networks are optimally structured for allometrically growing spatial regions. We show that isometrically- rather than allometrically-growing regions are most efficiently supplied, and our results shed light on two longstanding debates regarding the form of cardiovascular networks and river networks. We indicate the importance of how these examples differ fundamentally in terms of the dimensions of the networked spatial region and the ambient space. In particular, our results for cardiovascular networks imply that basal energy use for organisms scales optimally as a 2/3 power of body mass.
We show that the ideal scalings we derive are strongly supported empirically, and argue that any departures from them must be dictated by additional constraints.