ABSTRACT Using a scale-free N-body simulation generated with the abacusN-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 per cent or better between $∼\!\! 10^2$ and $10^5$ particles. For compaso, we find weaker convergence over a similar region, with at least 10 per cent between $∼\!\! 10^2$ and $10^4$ particles. Furthermore, we find that 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 per cent with as few as $∼\!\! 70$ particles for compaso and within 2 per cent for as few as $∼\!\! 30$ particles for rockstar.