Plants: The Bidirectional Optimizers

A tree optimizes in two directions simultaneously: upward toward light and downward toward water. The canopy and root system are not independent organs — they are coupled through Liebig's law of the minimum, where the limiting resource bottlenecks the entire organism. We introduce a coupled discrete-element model in which the tree allocates a fraction \(\alpha\) of its growth budget to canopy expansion and \((1 - \alpha)\) to root growth, with the budget itself determined by the minimum of total light energy and total water absorption. Simulating greedy-optimal growth across a grid of environments (light intensity \(\times\) water table depth), we find that the optimal allocation \(\alpha^\) varies smoothly from root-heavy (\(\alpha^ \approx 0.2\)) in dim environments to canopy-heavy (\(\alpha^ \approx 0.8\)) in bright environments with accessible water, shifting back toward balanced (\(\alpha^ \approx 0.5\)) when water is deep despite abundant light. The full tree morphology — crown shape, root architecture, shoot-to-root ratio, and size — is encoded by a five-dimensional environmental latent vector \(\boldsymbol{z} = (\theta, I_0, d, f_r, T)\). This paper unifies our companion studies on canopy architecture (Nagy, 2026a) and root architecture (Nagy, 2026b) into a single framework: the plant as a bidirectional optimizer whose form is decoded from the joint light-water environment.