Rémy's algorithm is a famous simple, yet efficient, recursive construction of uniform random binary plane trees. In this talk we will present a variation of this algorithm due to Bacher, Bodini, and Jacquot that we will exploit by taking its scaling limit in order to construct a continuous-time and continuous-space process that grows continuum random trees. This process can be defined with any initial tree; when started from the trivial singleton, at any positive time t, the tree is distributed as Aldous's Brownian CRT with mass t. This process also exhibits an ergodic property: When started from any initial distribution, the tree at time t, suitably rescaled to have constant mass, converges in distribution to the Brownian CRT.
This is based on a joint work with Nicolas Curien available at https://arxiv.org/pdf/2606.27035