  We consider a branching random walk (BRW) taking its values in the
{$\mathtt{b}$-ary rooted tree $\mathbb W_{ \mathtt{b}}$} (i.e. the set of
finite words written in the alphabet $\{ 1, \ldots, \mathtt{b} \}$, with
$\mathtt{b}\! \geq \! 2$). The BRW is indexed by a critical Galton--Watson tree
conditioned to have $n$ {vertices}; its offspring distribution is aperiodic and
is in the domain of attraction of a $\gamma$-stable law, $\gamma \in (1, 2]$.
The jumps of the BRW are those of a {nearest-neighbour} null-recurrent random
walk on $\mathbb W_{ \mathtt{b}}$ (reflection at the root of $\mathbb W_{
\mathtt{b}}$ and otherwise: probability $1/2$ to move closer to the root of
$\mathbb W_{ \mathtt{b}}$ and probability $1/(2\mathtt{b})$ to move away from
it to one of the $\mathtt{b}$ sites above). We denote by $\mathcal
R_{\mathtt{b}} (n)$ the range of the BRW in $\mathbb W_{ \mathtt{b}}$ which is
the set of all sites in $\mathbb W_{\mathtt{b}}$ visited by the BRW. We first
prove a law of large numbers for $\# \mathcal R_{\mathtt{b}} (n)$ and we also
{prove} that if we equip $\mathcal R_{\mathtt{b}} (n)$ (which is a random
subtree of $\mathbb W_{\mathtt{b}}$) with its graph-distance $d_{\mathtt{gr}}$,
then there exists a scaling sequence $(a_n)_{n\in \mathbb N}$ satisfying $a_n
\! \rightarrow \! \infty$ such that the metric space $(\mathcal R_{\mathtt{b}}
(n), a_n^{-1}d_{\mathtt{gr}})$, equipped with its normalised empirical measure,
converges to the reflected Brownian cactus with $\gamma$-stable branching
mechanism: namely, a random compact real tree that is a variant of the Brownian
cactus introduced by N. Curien, J-F. Le Gall and G. Miermont.
