  We analyse the cluster discovered by invasion percolation on a branching
process with a power-law offspring distribution. Invasion percolation is a
paradigm model of self-organised criticality, where criticality is approached
without tuning any parameter. By performing invasion percolation for $n$ steps,
and letting $n\to\infty$, we find an infinite subtree, called the invasion
percolation cluster (IPC). A notable feature of the IPC is its geometry that
consists of a unique path to infinity (also called the backbone) onto which
finite forests are attached. Our main theorem shows the volume scaling limit of
the $k$-cut IPC, which is the cluster containing the root when the edge between
the $k$-th and $(k+1)$-st backbone vertices is cut.
  We assume a power-law offspring distribution with exponent $\alpha$ and
analyse the IPC for different power-law regimes. In a finite-variance setting
$(\alpha>2)$ our results are a natural extension of previous works on the
branching process tree (Michelen et al. 2019) and the regular tree (Angel et
al. 2008). However, for an infinite-variance setting ($\alpha\in(1,2)$) or even
an infinite-mean setting ($\alpha\in(0,1)$), results significantly change. This
is illustrated by the volume scaling of the $k$-cut IPC, which scales as $k^2$
for $\alpha>2$, but as $k^{\alpha/(\alpha-1)}$ for $\alpha \in (1,2)$ and
exponentially for $\alpha \in (0,1)$.
