In their 1971 study of telephone switching circuitry, Graham and Pollak designed a novel addressing scheme that was better suited for the faster communication required by computers. They introduced the distance matrix of a graph, and used its eigenvalues to bound the size of the addresses in their scheme. We continue to study distance matrices coming from trees, and use their spectral properties to obtain new inequalities for trees, matroids, and valuated matroids. This is joint work with Graham Denham, Chris Eur, June Huh, and Botong Wang.