  A Brownian spatial tree is defined to be a pair $(\mathcal{T},\phi)$, where
$\mathcal{T}$ is the rooted real tree naturally associated with a Brownian
excursion and $\phi$ is a random continuous function from $\mathcal{T}$ into
$\mathbb{R}^d$ such that, conditional on $\mathcal{T}$, $\phi$ maps each arc of
$\mathcal{T}$ to the image of a Brownian motion path in $\mathbb{R}^d$ run for
a time equal to the arc length. It is shown that, in high dimensions, the
Hausdorff measure of arcs can be used to define an intrinsic metric
$d_{\mathcal{S}}$ on the set $\mathcal{S}:=\phi(\mathcal{T})$. Applications of
this result include the recovery of the spatial tree $(\mathcal{T},\phi)$ from
the set $\mathcal{S}$ alone, which implies in turn that a Dawson--Watanabe
super-process can be recovered from its range. Furthermore, $d_{\mathcal{S}}$
can be used to construct a Brownian motion on $\mathcal{S}$, which is proved to
be the scaling limit of simple random walks on related discrete structures. In
particular, a limiting result for the simple random walk on the branching
random walk is obtained.
