R-Trees and Symmetric Differences of Sets

Abstract

The supremum of the symmetric differencex▵y:= (x\\y)∪(y\\x) of subsetsx,yof R satisfies the so-called four-point condition; that is, for allx,x′,y,y′ ⊆ R, one has sup (x▵x′) + sup(y▵y′)≤ max { sup(x▵y) + sup(x′ ▵y′), sup (x▵y′) + sup(x′ ▵y)}. It follows that the setEof all subsets of R which are bounded from above forms a valuated matroid relative to the mapv:E×E→{-∞}∪R : (x,y)→sup (x▵y). Hence, according to T-theory, there exists an R-treeT(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 ofT(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 setBand an arbitrary mapr:B→{-∞}∪R, the map P(B) × P(B)→R∪{±∞}: (x,y)→supr(x▵y) again satisfies the four-point condition; so any non-empty setZof subsets ofBwith supr(x▵y)<∞ for allx,y∈Zforms 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→{-∞}∪R ,Z⊆ P(B)); consequently, since every R-tree can be embedded isometrically intoT(E,v)for some valuated matroid (E,v), every R-tree can, in principle, be described in terms of such a system.