Tropical geometry and Okounkov bodies are generalizations of the theory of Newton polytopes in different directs: tropical geometry for higher codimensions; Okounkov bodies for non-toric ambient spaces. In this talk, we will discuss joint work with Stefano Urbinati which finds these two theories converging again. We first discuss the analogue of Okounkov bodies over discrete valuation rings, using constructions motivated both by tropical geometry and Arakelov theory. In the special case of semistable families of curves, the theory of linear systems on graphs makes an appearance. This gives some pointers to a higher dimensional theory of combinatorial linear systems.