  We show the theory of pointed $\R$-trees with radius at most $r$ is
axiomatizable in a suitable continuous signature. We identify the model
companion $\rbRT_r$ of this theory and study its properties. In particular, the
model companion is complete and has quantifier elimination; it is stable but
not superstable. We identify its independence relation and find built-in
canonical bases for non-algebraic types. Among the models of $\rbRT_r$ are
$\R$-trees that arise naturally in geometric group theory. In every infinite
cardinal, we construct the maximum possible number of pairwise non-isomorphic
models of $\rbRT_r$; indeed, the models we construct are pairwise
non-homeomorphic. We give detailed information about the type spaces of
$\rbRT_r$. Among other things, we show that the space of $2$-types over the
empty set is nonseparable. Also, we characterize the principal types of finite
tuples (over the empty set) and use this information to conclude that $\rbRT_r$
has no atomic model.
