  The paper studies a class of variational problems, modeling optimal shapes
for tree roots. Given a measure $\mu$ describing the distribution of root hair
cells, we seek to maximize a harvest functional $\mathcal{H}$, computing the
total amount of water and nutrients gathered by the roots, subject to a cost
for transporting these nutrients from the roots to the trunk. Earlier papers
had established the existence of an optimal measure, and a priori bounds. Here
we derive necessary conditions for optimality. Moreover, in space dimension
$d=2$, we prove that the support of an optimal measure is nowhere dense.
