In this paper we study distributive lattices in the framework of Formal Concept Analysis (FCA). The main motivation comes from phylogeny where biological derivations and parsimonious trees can be represented as median graphs. There exists a close connection between distributive lattices and median graphs. Moreover, FCA provides efficient algorithms to build concept lattices. However, a concept lattice is not necessarily distributive and thus it is not necessarily a median graph. In this paper we investigate possible ways of transforming a concept lattice into a distributive one, by making use Birkhoff’s representation of distributive lattices. We detail the operation that transforms a reduced context into a context of a distributive lattice. This allows us to reuse the FCA algorithmic machinery to build and to visualize distributive concept lattices, and then to study the associated median graphs.