  We consider so-called discrete snakes obtained from size-conditioned critical
Bienaym\'e-Galton-Watson trees by assigning to each node a random spatial
position in such a way that the increments along each edge are i.i.d. When the
offspring distribution belongs to the domain of attraction of a stable law with
index $\alpha \in (1,2]$, we give a necessary and sufficient condition on the
tail distribution of the spatial increments for this spatial tree to converge,
in a functional sense, towards the Brownian snake driven by the $\alpha$-stable
L\'evy tree. We also study the case of heavier tails, and apply our result to
study the number of inversions of a uniformly random permutation indexed by the
tree.
