In this talk, I will give an introduction to the theory of
ramified optimal transportation. In terms of applied mathematics,
transport paths are used to model many "tree shaped" branching
structures, which are commonly found in many living and nonliving
systems. Trees, lungs, river channel networks, are just some
examples. On the other hand, optimal transport paths provide
excellent examples for studying geodesic problems in generalized
metric spaces, where the distance functions do not necessarily
satisfy the usual triangle inequality, but satisfy a relaxed
triangle inequality. In the end, we will use the theory to explain
the dynamic formation of tree leaves. We will see how tree leaves
grow beautiful shapes and vein patterns in nature.