  The real trees form a class of metric spaces that extends the class of trees
with edge lengths by allowing behavior such as infinite total edge length and
vertices with infinite branching degree. Aldous's Brownian continuum random
tree, the random tree-like object naturally associated with a standard Brownian
excursion, may be thought of as a random compact real tree. The continuum
random tree is a scaling limit as N tends to infinity of both a critical
Galton-Watson tree conditioned to have total population size N as well as a
uniform random rooted combinatorial tree with N vertices. The Aldous--Broder
algorithm is a Markov chain on the space of rooted combinatorial trees with N
vertices that has the uniform tree as its stationary distribution. We construct
and study a Markov process on the space of all rooted compact real trees that
has the continuum random tree as its stationary distribution and arises as the
scaling limit as N tends to infinity of the Aldous--Broder chain. A key
technical ingredient in this work is the use of a pointed Gromov--Hausdorff
distance to metrize the space of rooted compact real trees.
