Editing Real tree (section)

== Characterizations ==
[[File:Four_point_condition.png|thumb|260x260px|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 [[Tree (data structure)|trees]] as graphs)'' A real tree is a [[Intrinsic_metric|geodesic]] [[metric space]] which contains no subset [[Homeomorphism|homeomorphic]] to a circle.<ref>{{Cite book |last=Chiswell |first=Ian |url=https://www.worldcat.org/oclc/268962256 |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 [[Δ-hyperbolic space|0-hyperbolic]] 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]] 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 [[Tree (graph theory)#Plane tree|plane trees]] by their [[Galton-Watson tree|contour process]]).'' 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 [[Metric space|pseudometric]] and an [[equivalence relation]] with: 

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

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

Then, the [[Quotient space (topology)|quotient space]] <math>([a,b]/\sim_e\, ,\, d_e) </math> is a real tree.<ref name=":0" />   Intuitively, the [[local minima]] of the excursion ''e'' are the parents of the [[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).

[[File:Collage.ogg|center|thumb|300x300px|Partant d'une [[Excursion brownienne|excursion]] ''e'' (en noir), la déformation (en vert) représente le « pliage » de la courbe jusqu'au « collage » des points d'une même classe d'équivalence, l'état final est l'arbre réel associé à ''e''.]]