The structure of the 3-separations of 3-connected matroids II

The authors showed in an earlier paper that there is a tree that displays, up to a natural equivalence, all non-trivial 3-separations of a 3-connected matroid. The purpose of this paper is to show that if certain natural conditions are imposed on the tree, then it has a uniqueness property. In particular, suppose that, from every pair of edges that meet at a degree-2 vertex and have their other ends of degree at least three, one edge is contracted. Then the resulting tree is unique.