Living organisms come in an immense variety of shapes, such as roots, branches, leaves and flowers in plants, or bones in animals. In some cases it is expected that, through natural selection, these organisms might have evolved into a ``best possible" shape. From a mathematical point of view, it is of interest to identify functionals whose minimizers mimic some of the shapes found in the biological world.
In this talk I shall discuss three functionals, modeling (i) the total amount of sunlight captured by the leaves of a tree, (ii) the amount of nutrients collected by the roots, and (iii) a ``ramified transportation cost", for transporting nutrients from the roots to the base of the trunk, or from the base of the trunk to the leaves.
The talk will address the existence and properties of optimal solutions. Recent work has also focused on the approximation of the ramified transport cost with more regular functionals, based on Gamma-convergence. This leads to new computational algorithms.