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The roots of a plant can be viewed as a graph, with vertices representing points in Euclidean space and edges representing the connective material. Such graphs will ideally be designed in order to optimize one or more relevant objectives. Two objectives that are particularly important for biological reasons include minimizing the total length of the edges in the graph (wiring cost), and minimizing the total lengths of the shortest paths from each point to the root of the graph (conduction delay). These two objectives compete with each other, as optimizing for one objective generally results in poorer performance of the other. Additionally, sensitivity to gravitational forces constrains the curvature of edges in the graph, and by extension how well these objectives can be optimized. In this paper we show how ideas from economics can be used to resolve the tradeoff between competing objectives, and we define a concrete optimization problem that accounts for the role of gravity in constraining the solution space. We then show two techniques for solving this seemingly impossible optimization problem, both of which will be accessible to undergraduate math students. 

Creating a routing backbone is a fundamental problem in both biology and engineering. The routing backbone of the trail networks of arboreal turtle ants (Cephalotes goniodontus) connects many nests and food sources using trail pheromone deposited by ants as they walk. Unlike species that forage on the ground, the trail networks of arboreal ants are constrained by the vegetation. We examined what objectives the trail networks meet by comparing the observed ant trail networks with networks of random, hypothetical trail networks in the same surrounding vegetation and with trails optimized for four objectives: minimizing path length, minimizing average edge length, minimizing number of nodes, and minimizing opportunities to get lost. The ants’ trails minimized path length by minimizing the number of nodes traversed rather than choosing short edges. In addition, the ants’ trails reduced the opportunity for ants to get lost at each node, favoring nodes with 3D configurations most likely to be reinforced by pheromone. Thus, rather than finding the shortest edges, turtle ant trail networks take advantage of natural variation in the environment to favor coherence, keeping the ants together on the trails.