Formal definition

A triangle in a real tree

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

Simple examples

• If ${\displaystyle X}$ 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 cycles). Such a tree is often called a simplicial tree. They are characterised by the following topological property: a real tree ${\displaystyle T}$ is simplicial if and only if the set of singular points of ${\displaystyle X}$ (points whose complement in ${\displaystyle X}$ has three or more connected components) is closed and discrete in ${\displaystyle X}$.
• The ${\displaystyle \mathbb {R} }$-tree obtained in the following way is nonsimplicial. Start with the interval [0, 2] and glue, for each positive integer n, an interval of length 1/n to the point 1 − 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 ${\displaystyle \mathbb {R} }$-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 ${\displaystyle P}$, and if two points are on the same ray from ${\displaystyle P}$, 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 ${\displaystyle P}$.
• 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.

Brownian trees

A Brownian tree^[4] is a stochastic process whose value is a (non-simplicial) real tree almost surely. Brownian trees arise as limits of various random processes on finite trees.^[5]

Ultralimits of metric spaces

Any ultralimit of a sequence ${\displaystyle (X_{i})}$ of ${\displaystyle \delta _{i}}$-hyperbolic spaces with ${\displaystyle \delta _{i}\to 0}$ is a real tree. In particular, the asymptotic cone of any hyperbolic space is a real tree.

Limit of group actions

Let ${\displaystyle G}$ be a group. For a sequence of based ${\displaystyle G}$-spaces ${\displaystyle (X_{i},*_{i},\rho _{i})}$ there is a notion of convergence to a based ${\displaystyle G}$-space ${\displaystyle (X_{\infty },x_{\infty },\rho _{\infty })}$ due to M. Bestvina and F. Paulin. When the spaces are hyperbolic and the actions are unbounded the limit (if it exists) is a real tree.^[6]

A simple example is obtained by taking ${\displaystyle G=\pi _{1}(S)}$ where ${\displaystyle S}$ is a compact surface, and ${\displaystyle X_{i}}$ the universal cover of ${\displaystyle S}$ with the metric ${\displaystyle i\rho }$ (where ${\displaystyle \rho }$ is a fixed hyperbolic metric on ${\displaystyle S}$).

This is useful to produce actions of hyperbolic groups on real trees. Such actions are analyzed using the so-called Rips machine. A case of particular interest is the study of degeneration of groups acting properly discontinuously on a real hyperbolic space (this predates Rips', Bestvina's and Paulin's work and is due to J. Morgan and P. Shalen^[7]).

Algebraic groups

If ${\displaystyle F}$ is a field with an ultrametric valuation then the Bruhat–Tits building of ${\displaystyle \mathrm {SL} _{2}(F)}$ is a real tree. It is simplicial if and only if the valuations is discrete.

${\displaystyle \Lambda }$-trees

If ${\displaystyle \Lambda }$ is a totally ordered abelian group there is a natural notion of a distance with values in ${\displaystyle \Lambda }$ (classical metric spaces correspond to ${\displaystyle \Lambda =\mathbb {R} }$). There is a notion of ${\displaystyle \Lambda }$-tree^[8] which recovers simplicial trees when ${\displaystyle \Lambda =\mathbb {Z} }$ and real trees when ${\displaystyle \Lambda =\mathbb {R} }$. The structure of finitely presented groups acting freely on ${\displaystyle \Lambda }$-trees was described. ^[9] In particular, such a group acts freely on some ${\displaystyle \mathbb {R} ^{n}}$-tree.

Real buildings

The axioms for a building can be generalized to give a definition of a real building. These arise for example as asymptotic cones of higher-rank symmetric spaces or as Bruhat-Tits buildings of higher-rank groups over valued fields.