Tangles, trees, and flowers

A tangle of order k in a matroid or graph may be thought of as a “k-connected component”. For a tangle of order k in a matroid or graph that satisfies a certain robustness condition, we describe a tree decomposition of the matroid or graph that displays, up to a certain natural equivalence, all the k-separations of the matroid or graph that are non-trivial with respect to the tangle.