# Real tree

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In [mathematics](/source/mathematics), '''real trees''' (also called '''<math>\mathbb R</math>-trees''') are a class of [metric space](/source/metric_space)s generalising simplicial [trees](/source/Tree_(graph_theory)). They arise naturally in many mathematical contexts, in particular [geometric group theory](/source/geometric_group_theory) and [probability theory](/source/probability_theory). They are also the simplest examples of [Gromov hyperbolic space](/source/Gromov_hyperbolic_space)s.

== Definition and examples ==

=== Formal definition ===

130px|thumb|class=skin-invert-image|A triangle in a real tree

A metric space <math>X</math> is a real tree if it is a [geodesic space](/source/Geodesic_metric_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 [topological](/source/topology) 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](/source/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>).

=== 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 [cycles](/source/Cycle_(graph_theory))). 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](/source/closed_set) of singular points at the expense of discreteness.
* The [Paris metric](/source/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](/source/hedgehog_space), a collection of line segments joined at a common endpoint. Any such space is a real tree.

== Characterizations ==
thumb|260x260px|class=skin-invert-image|Visualisation of the four points condition and the 0-hyperbolicity. In green: <math>(x,y)_t=(y,z)_t</math> ; in blue: <math>(x,z)_t</math>.
Here are equivalent characterizations of real trees which can be used as definitions:

1) ''(similar to [trees](/source/Tree_(data_structure)) as graphs)'' A real tree is a [geodesic](/source/Intrinsic_metric) [metric space](/source/metric_space) which contains no subset [homeomorphic](/source/Homeomorphism) to a circle.<ref>{{Cite book |last=Chiswell |first=Ian |title=Introduction to [lambda]-trees |date=2001 |publisher=World Scientific |isbn=978-981-281-053-3 |location=Singapore |oclc=268962256}}</ref>

2) A real tree is a connected metric space <math>(X,d)</math> which has the '''four points condition'''<ref>Peter Buneman, ''A Note on the Metric Properties of Trees'', Journal of combinatorial theory, B (17), {{p.|48-50}}, 1974.</ref> (see figure): 
:For all <math>x,y,z,t\in X,</math>  <math> d(x,y)+d(z,t)\leq \max[d(x,z)+d(y,t)\,;\, d(x,t)+d(y,z)]</math>.

3) A real tree is a connected [0-hyperbolic](/source/%CE%94-hyperbolic_space) metric space<ref name=":0">{{Cite book |last=Evans |first=Stevan N. |title=Probability and Real Trees |publisher=École d’Eté de Probabilités de Saint-Flour XXXV |year=2005}}</ref> (see figure). Formally, 
:For all <math>x,y,z,t\in X,</math>  <math> (x,y)_t\geq \min [ (x,z)_t\, ; \, (y,z)_t ],</math>
where <math>(x,y)_t</math> denotes the [Gromov product](/source/Gromov_product) of <math>x</math> and <math>y</math> with respect to <math>t</math>, that is, <math>\textstyle\frac 1 2 \left( d(x, t) + d(y, t) - d(x, y) \right).</math>

4) ''(similar to the characterization of [plane trees](/source/Tree_(graph_theory)) by their [contour process](/source/Galton-Watson_tree)).'' Consider a positive excursion of a function. In other words, let <math>e</math> be a continuous real-valued function and <math>[a,b]</math> an interval such that <math>e(a)=e(b)=0</math> and <math>e(t)>0</math> for <math>t\in ]a,b[</math>.

For <math>x, y\in [a,b]</math>, <math>x\leq y</math>, define a [pseudometric](/source/Metric_space) and an [equivalence relation](/source/equivalence_relation) with: 

<math display=block> d_e( x, y) := e(x)+e(y)-2\min(e(z)\, ;z\in[x,y]),</math> 

<math display=block> x\sim_e y \Leftrightarrow d_e(x,y)=0.</math>  

Then, the [quotient space](/source/Quotient_space_(topology)) <math>([a,b]/\sim_e\, ,\, d_e) </math> is a real tree.<ref name=":0" />   Intuitively, the [local minima](/source/local_minima) of the excursion ''e'' are the parents of the [local maxima](/source/local_maxima). Another visual way to construct the real tree from an excursion is to "put glue" under the curve of ''e'', and "bend" this curve, identifying the glued points (see animation).

center|thumb|300x300px

== Examples ==

Real trees often appear, in various situations, as limits of more classical metric spaces.

=== Brownian trees ===

A [Brownian tree](/source/Brownian_tree)<ref>{{citation | last = Aldous | first = D. | author-link = David Aldous | date = 1991 | title = The continuum random tree I | journal = [Annals of Probability](/source/Annals_of_Probability) | volume = 19 | pages = 1–28| doi = 10.1214/aop/1176990534 | doi-access = free }}</ref> is a random metric space whose value is a (non-simplicial) real tree almost surely. Brownian trees arise as limits of various random processes on finite trees.<ref>{{citation | last = Aldous | first = D. | author-link = David Aldous | date = 1991 | title = The continuum random tree III | journal = [Annals of Probability](/source/Annals_of_Probability) | volume = 21 | pages = 248–289}}</ref>

=== Ultralimits of metric spaces ===

Any [ultralimit](/source/ultralimit) of a sequence <math>(X_i)</math> of <math>\delta_i</math>-[hyperbolic](/source/Hyperbolic_metric_space) spaces with <math>\delta_i \to 0</math> is a real tree. In particular, the [asymptotic cone](/source/Ultralimit) of any hyperbolic space is a real tree.

=== Limit of group actions ===

Let <math>G</math> be a [group](/source/group_(mathematics)). For a sequence of based <math>G</math>-spaces <math>(X_i, *_i, \rho_i)</math> there is a notion of convergence to a based <math>G</math>-space <math>(X_\infty, x_\infty, \rho_\infty)</math> 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.<ref>{{citation | last = Bestvina | first = Mladen | author-link = Mladen Bestvina | title = Handbook of Geometric Topology | contribution = <math>\mathbb R</math>-trees in topology, geometry and group theory | pages = 55–91 | year = 2002 | publisher = Elsevier | isbn = 9780080532851 | url = https://books.google.com/books?id=8OYxdADnhZoC&pg=PA55}}</ref>

A simple example is obtained by taking <math>G = \pi_1(S)</math> where <math>S</math> is a [compact](/source/compact_space) surface, and <math>X_i</math> the universal cover of <math>S</math> with the metric <math>i\rho</math> (where <math>\rho</math> is a fixed hyperbolic metric on <math>S</math>).

This is useful to produce actions of hyperbolic groups on real trees. Such actions are analyzed using the so-called [Rips machine](/source/Rips_machine). A case of particular interest is the study of degeneration of groups acting [properly discontinuously](/source/Group_action) on a [real hyperbolic space](/source/Hyperbolic_space) (this predates Rips', Bestvina's and Paulin's work and is due to J. Morgan and [P. Shalen](/source/Peter_Shalen)<ref>{{citation
 | last = Shalen | first = Peter B. | author-link = Peter Shalen
 | editor-last = Gersten | editor-first = S. M.
 | contribution = Dendrology of groups: an introduction
 | isbn = 978-0-387-96618-2
 | mr = 919830
 | pages = 265–319
 | publisher = [Springer-Verlag](/source/Springer-Verlag)
 | series = Math. Sci. Res. Inst. Publ.
 | title = Essays in Group Theory
 | volume = 8
 | year = 1987}}</ref>).

=== Algebraic groups ===

If <math>F</math> is a [field](/source/field_(mathematics)) with an [ultrametric](/source/ultrametric_space) [valuation](/source/Valuation_(algebra)) then the [Bruhat–Tits building](/source/Building_(mathematics)) of <math>\mathrm{SL}_2(F)</math> is a real tree. It is simplicial if and only if the valuations is discrete.

== Generalisations ==

=== Λ-trees ===

If <math>\Lambda</math> is a [totally ordered abelian group](/source/totally_ordered_abelian_group) there is a natural notion of a distance with values in <math>\Lambda</math> (classical metric spaces correspond to <math>\Lambda = \mathbb R</math>). There is a notion of <math>\Lambda</math>-tree<ref>{{citation | last = Chiswell | first = Ian
 | isbn = 981-02-4386-3
 | location = River Edge, NJ
 | mr = 1851337
 | publisher = World Scientific Publishing Co. Inc.
 | title = Introduction to Λ-trees
 | year = 2001}}</ref> which recovers simplicial trees when <math>\Lambda = \mathbb Z</math> and real trees when <math>\Lambda = \mathbb R</math>. The structure of [finitely presented group](/source/finitely_presented_group)s acting [freely](/source/Group_action) on <math>\Lambda</math>-trees was described. <ref>{{citation | last = O. Kharlampovich, A. Myasnikov, D. Serbin | title = Actions, length functions and non-archimedean words IJAC 23, No. 2, 2013.}}</ref> In particular, such a group acts freely on some  <math>\mathbb R^n</math>-tree.

=== Real buildings ===

The axioms for a [building](/source/Building_(mathematics)) can be generalized to give a definition of a real building. These arise for example as asymptotic cones of higher-rank [symmetric spaces](/source/symmetric_spaces) or as Bruhat-Tits buildings of higher-rank groups over valued fields.

==See also==
*[Dendroid (topology)](/source/Dendroid_(topology))
*[Tree-graded space](/source/Tree-graded_space)

== References ==

{{reflist}}

Category:Group theory
Category:Geometry
Category:Topology
Category:Trees (topology)

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Adapted from the Wikipedia article [Real tree](https://en.wikipedia.org/wiki/Real_tree) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Real_tree?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.