Matroid tree-width

We show that the tree-width of a graph can be defined without reference to graph vertices, and hence the notion of tree-width can be naturally extended to matroids. (This extension was inspired by an original unpublished idea of Jim Geelen.) We prove that the tree-width of a graphic matroid is equal to that of its underlying graph. Furthermore, we extend the well-known relation between the branch-width and the tree-width of a graph to all matroids.