Real Tree

In mathematics, a real tree, or an -tree, is a metric space (M,d) such that for any x, y in M there is a unique arc from x to y and this arc is a geodesic segment. Here by an arc from x to y we mean the image in M of a topological embedding f from an interval to M such that f(a)=x and f(b)=y. The condition that the arc is a geodesic segment means that the map f above can be chosen to be an isometric embedding, that is it can be chosen so that for every z, t in we have d(f(z), f(t))=|z-t| and that f(a)=x, f(b)=y.

Equivalently, a geodesic metric space M is a real tree if and only if M is a δ-hyperbolic space with δ=0.

Complete real trees are injective metric spaces (Kirk 1998).

There is a theory of group actions on R-trees, known as the Rips machine, which is part of geometric group theory.