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.
Read more about Real Tree: Simplicial R-trees, Examples
Famous quotes containing the words real and/or tree:
“To die proudly when it is no longer possible to live proudly. Death freely chosen, death at the right time, brightly and cheerfully accomplished amid children and witnesses: then a real farewell is still possible, as the one who is taking leave is still there; also a real estimate of what one has wished, drawing the sum of one’s life—all in opposition to the wretched and revolting comedy that Christianity has made of the hour of death.”
—Friedrich Nietzsche (1844–1900)
“Sir, he throws away his money without thought and without merit. I do not call a tree generous that sheds its fruit at every breeze.”
—Samuel Johnson (1709–1784)