Emergent Root Architecture from Discrete Tip Dynamics: The Underground Mirror

This paper is the underground twin of "Emergent Canopy Architecture from Discrete Leaf Dynamics" (Nagy, 2026). Where the canopy paper showed that tree crown shape emerges from the interaction of leaf placement and light-driven die-off, we show that root system architecture emerges from a symmetric but fundamentally different process: root tip placement and water-depletion-driven die-off. The key physical difference is that canopy competition is directional (leaves above block light below) while root competition is isotropic (each root tip depletes water in all directions). This single difference explains why canopies form hollow shells while root systems fill volumes. We simulate six root strategies (taproot, fibrous, lateral, deep-spread, surface-mat, sinker) and a greedy-optimal strategy, showing that the optimal root architecture is a smooth function of the water table depth \(d\), rain frequency \(f_r\), and tree age \(T\). The environmental latent vector \(\boldsymbol{z}_{\text{root}} = (d, f_r, T)\) mirrors the canopy's \(\boldsymbol{z}_{\text{canopy}} = (\theta, E, T)\), completing a symmetric picture: the above-ground form is decoded from the light environment, the below-ground form from the water environment, and together they constitute the tree's full morphological latent representation.