  We develop an excursion theory for Brownian motion indexed by the Brownian
tree, which in many respects is analogous to the classical It\^o theory for
linear Brownian motion. Each excursion is associated with a connected component
of the complement of the zero set of the tree-indexed Brownian motion. Each
such connectedcomponent is itself a continuous tree, and we introduce a
quantity measuring the length of its boundary. The collection of boundary
lengths coincides with the collection of jumps of a continuous-state branching
process with branching mechanism $\psi(u)=\sqrt{8/3}\,u^{3/2}$. Furthermore,
conditionally on the boundary lengths, the different excursions are
independent, and we determine their conditional distribution in terms of an
excursion measure $\mathbb{M}_0$ which is the analog of the It\^o measure of
Brownian excursions. We provide various descriptions of the excursion measure
$\mathbb{M}_0$, and we also determine several explicit distributions, such as
the joint distribution of the boundary length and the mass of an excursion
under $\mathbb{M}_0$. We use the Brownian snake as a convenient tool for
defining and analysing the excursions of our tree-indexed Brownian motion.
