In nature, many transport systems have ramified (i.e. branching) structures.Trees, railways, airlines, lightning, electric power supply, the circulatory system, the river channel networks, and cardiovascular systems are just some examples. In this talk, I will talk about how to set up a mathematical theory for this general phenomenon in terms of optimal transport paths. After studying some nice properties of optimal transport paths, we will also discuss the connection of the theory with the Monge-Kantorovich transport problem. In the end, I will provide the description of the dynamic formation of a tree leaf, as an application of this theory.