The thermal design of natural convection heat sinks is critical to the heat management of electronic devices. In this paper, topology optimization (TO) with a new parameter advancing scheme is employed to study the effect of the heating power, heat source size, allowed volume and material thermal conductivity on the optimized design of natural convection heat sinks. Except for the heating power, the other three factors also play an important role on the optimization results. For the small heating power, TO predicts the tree-like heat sink with many secondary-branches to improve heat conduction, but the specific structure is highly dependent on the heat source size, allowed volume and material thermal conductivity. For the large heating power, TO prefers to produce the taper-like heat sink where the tiny branches fade away to adapt to the strong flow, and the influences of the other three factors are reduced. However, two primary branches that connect the heat source and top corners of the design domain always exist, indicating that they are the most effective ways to improve heat transfer. This work has highlighted the impact of different parameters in TO of natural convection heat sinks and provides a more in-depth understanding of the design guidelines. [ABSTRACT FROM AUTHOR]