  We consider the class of graphs for which the edge connectivity is equal to
the maximum number of edge-disjoint spanning trees, and the natural
generalization to matroids, where the cogirth is equal to the number of
disjoint bases. We provide descriptions of such graphs and matroids, showing
that such a graph (or matroid) has a unique decomposition. In the case of
graphs, our results are relevant for certain communication protocols.
