  Using a scale-free $N$-body simulation generated with the ABACUS $N$-body
code, we test the robustness of halo mass accretion histories via their
convergence to self-similarity. We compare two halo finders, ROCKSTAR and
COMPASO. We find superior self-similarity in halo mass accretion histories
determined using ROCKSTAR, with convergence to 5% or better between $\sim10^2$
to $10^5$ particles. For COMPASO we find weaker convergence over a similar
region, with at least 10% between $\sim10^2$ to $10^4$ particles. Furthermore,
we find the convergence to self-similarity improves as the simulation evolves,
with the largest and deepest regions of convergence appearing after the scale
factor quadrupled from the time at which non-linear structures begin to form.
With sufficient time evolution, halo mass accretion histories are converged to
self-similarity within 5% with as few as $\sim70$ particles for COMPASO and
within 2% for as few as $\sim30$ particles for ROCKSTAR.
