Editing Brownian tree (section)

=== Continuous tree ===
The Brownian tree is a [[real tree]] defined from a [[Brownian excursion]] (see characterisation 4 in [[Real tree]]).

Let <math>e=(e(x),0\leq x\leq 1)</math>be a Brownian excursion. Define a [[Metric space|pseudometric]] <math>d</math> on <math>[0,1]</math> with

: <math> d(x, y) := e(x)+e(y)-2 \min\big\{e(z)\, ; z\in[x,y]\big\}, </math> for any <math>x,y\in [0,1]</math>

We then define an [[equivalence relation]], noted <math>\sim_e</math> on <math>[0,1]</math> which relates all points <math>x,y</math> such that <math>d(x,y)=0</math> .

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

<math>d</math> is then a distance on the [[Quotient space (topology)|quotient space]] <math>[0,1]\,/\!\sim_e</math>. 
{{Math theorem
| math_statement = The random metric space <math>\big([0,1]\,/\!\sim_e,\, d\big)</math> is called a '''Brownian tree'''.
| name = Definition
}}
It is customary to consider the excursion <math>e/2</math> rather than <math>e</math>.