Oxley, J.Semple, C.Whittle, G.2008-10-162008-10-162007Oxley, J., Semple, C., Whittle, G. (2007) The structure of the 3-separations of 3-connected matroids II. European Journal of Combinatorics, 28(4), pp. 1239-1261.http://hdl.handle.net/10092/1704The 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.enJournal ArticleFields of Research::230000 Mathematical Sciences::230100 Mathematics::230103 Rings and algebrasFields of Research::230000 Mathematical Sciences::230100 Mathematics::230101 Mathematical logic, set theory, lattices and combinatoricshttps://doi.org/10.1016/j.ejc.2006.01.007