Considering the increasing gap between energy demand and clean energy supply availability, there is a strong need to impart novel design structures to effectively utilize the available resources and energy storage systems. Nature-inspired fin structures embedded with well-chosen phase change materials (PCM) can play an essential role in improving the heat transfer in thermal energy storage devices. The present work investigates the melting and solidification processes in a metallic heat exchanger filled with n-octadecane as working PCM, with tree-shaped fins. The study is conducted for various fin generation numbers, volume fractions, length ratios, and angles between the Y-shaped tree fin branches. The results are discussed in the form of melt fraction of PCM, heated surface average temperature variation, and thermal energy stored. The results clearly indicate that the heat transfer is significantly improved with increasing branches of the tree-structured fins because of an increased area available for heat transfer. This study can be utilized to enhance the performance of PCM embedded thermal energy storage devices.