Editing Real tree (section)

=== Simple examples ===

*If <math>X</math> is a connected graph with the combinatorial metric then it is a real tree if and only if it is a tree (i.e. it has no [[Cycle (graph theory)|cycles]]). Such a tree is often called a simplicial tree. They are characterised by the following topological property: a real tree <math>T</math> is simplicial if and only if the set of singular points of <math>X</math> (points whose complement in <math>X</math> has three or more connected components) is closed and discrete in <math>X</math>.
* The <math>\mathbb R</math>-tree obtained in the following way is nonsimplicial. Start with the interval [0,&thinsp;2] and glue, for each positive integer ''n'', an interval of length 1/''n'' to the point 1&thinsp;−&thinsp;1/''n'' in the original interval. The set of singular points is discrete, but fails to be closed since 1 is an ordinary point in this <math>\mathbb R</math>-tree. Gluing an interval to 1 would result in a [[closed set]] of singular points at the expense of discreteness.
* The [[Paris metric]] makes the plane into a real tree. It is defined as follows: one fixes an origin <math>P</math>, and if two points are on the same ray from <math>P</math>, their distance is defined as the Euclidean distance. Otherwise, their distance is defined to be the sum of the Euclidean distances of these two points to the origin <math>P</math>.
* The plane under the Paris metric is an example of a [[hedgehog space]], a collection of line segments joined at a common endpoint. Any such space is a real tree.