The supremum of the symmetric difference x y := (x \ y) βͺ (y \ x) of subsets x, y of R satisfies the so-called four-point condition; that is, for all x, x , y, y β R, one has
It follows that the set E of all subsets of R which are bounded from above forms a valuated matroid relative to the map v:
Hence, according to T-theory, there exists an R-tree T (E,v) uniquely determined by (E, v) up to isometry, the ends of which correspond in a one-to-one fashion to the elements of the completion of (E, v). In addition, the points of T (E,v) can be identified with those bounded subsets of R which contain their infimum.
Here, we show that these observations hold true in a much more general setting: given an arbitrary non-empty set B and an arbitrary map r : B β {-β} βͺ R, the map P(B) Γ P(B) β R βͺ {Β±β} : (x, y) β sup r(x y) again satisfies the four-point condition; so any non-empty set Z of subsets of B with sup r(x y) < β for all x, y β Z forms a valuated matroid of rank 2 relative to this map and, therefore, gives rise to an R-tree.
It is shown here that every valuated matroid of rank 2 can be realized in this way by choosing an appropriate system (B, r : B β {-β} βͺ R, Z β P(B)); consequently, since every R-tree can be embedded isometrically into T (E,v) for some valuated matroid (E, v), every R-tree can, in principle, be described in terms of such a system.