In this note, we provide a new characterization of Aldous' Brownian continuum random tree as the unique fixed point of a certain natural operation on continuum trees (which gives rise to a recursive distributional equation). We also show that this fixed point is attractive.
"The Brownian continuum random tree as the unique solution to a fixed point equation." Electron. Commun. Probab. 20 1 - 14, 2015. https://doi.org/10.1214/ECP.v20-4250