A Unified Scaling Law for Biological Transport Networks: Alveoli and Plant Branching Systems

Biological systems exhibit convergent structural organization across kingdoms despite divergent physiological roles. Here we show that pulmonary alveoli and plant branching networks obey a unified scaling law governed by boundary-mediated growth and surface-constrained transport. A generalized growth relation of the form dM/dt = C Mα, with α = (d−1+β)/d, provides a common framework for classifying biological transport systems. Pulmonary alveolar structures operate near the geometric baseline (α = 2/3, β ≈ 0), consistent with diffusion-limited exchange at the alveolar–capillary interface. In contrast, plant vascular networks exhibit transport amplification (β > 0), yielding α ≈ 3/4, consistent with hierarchical, energy-efficient transport. These results demonstrate a structural equivalence between respiratory and plant transport systems and support a unified boundary-mediated scaling principle governing biological transport architectures. The classification applies at the functional exchange boundary where flux is realized.