  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.
