Editing Real tree (section)

=== Formal definition ===

[[File:Y property.png|130px|thumb|A triangle in a real tree]]

A metric space <math>X</math> is a real tree if it is a [[Geodesic metric space|geodesic space]] where every triangle is a tripod. That is, for every three points <math>x, y, \rho \in X</math> there exists a point <math>c = x \wedge y</math> such that the geodesic segments <math>[\rho,x], [\rho,y]</math> intersect in the segment <math>[\rho,c]</math> and also <math>c \in [x,y]</math>. This definition is equivalent to <math>X</math> being a "zero-hyperbolic space" in the sense of Gromov (all triangles are "zero-thin"). 
Real trees can also be characterised by a [[topology|topological]] property. A metric space <math>X</math> is a real tree if for any pair of points <math>x, y \in X</math> all [[topological embedding]]s <math>\sigma</math> of the segment <math>[0,1]</math> into <math>X</math> such that <math>\sigma(0) = x, \, \sigma(1) = y</math> have the same image (which is then a geodesic segment from <math>x</math> to <math>y</math>).