The performance enhancement of latent heat storage (LHS) units is of great consequence for the development of sustainable energy. In this article, the transient models of phase-change heat transfer processes in LHS units with consideration of natural convection are developed and numerically analyzed, in an effort to explore the role of the fractal tree-shaped fin in the energy charging performance. The solid-liquid interface evolution and dynamic temperature response in the tree-shaped finned LHS unit are presented and compared with the wheel-shaped one. Moreover, the influence of the fin number is investigated for maximizing the energy charging performance. The results indicate that the tree-shaped fin has merits of effective layout optimization for the point-to-area heat transfer and the time-coordination of energy charging and discharging process. Compared with the wheel-shaped fin, the melting duration time is decreased only a little for the presence of tree-shaped fin, however, the solidification process is decreased significantly. The presence of tree-shaped fin leads to a lower energy charging rate during the early and middle stages of the charging process due to the natural convection suppression, however, the energy charging rate is faster during the later stage due to the thermal conduction dominated interior heat transfer. For maximizing the melting performance, the appropriate fin number in practical applications is 16.