  We use Dirichlet form methods to construct and analyze a reversible Markov
process, the stationary distribution of which is the Brownian continuum random
tree. This process is inspired by the subtree prune and regraft (SPR) Markov
chains that appear in phylogenetic analysis. A key technical ingredient in this
work is the use of a novel Gromov--Hausdorff type distance to metrize the space
whose elements are compact real trees equipped with a probability measure.
Also, the investigation of the Dirichlet form hinges on a new path
decomposition of the Brownian excursion.
