Brownian Continuum Random Tree

The Brownian continuum random tree is the random metric measure real tree coded by a normalized Brownian excursion on . For , define

Identifying points at zero distance gives the real tree, rooted at the class of 0. Its probability measure is the pushforward measure of Lebesgue measure on under the map sending each to its equivalence class. The excursion normalization fixes the multiplicative convention for the metric. Aldous (1993, §2.7) treats the excursion construction, using twice a normalized Brownian excursion for his metric convention.