The behaviors of thin indium films in the early stage of growth are studied in situ by a novel ultra-sensitive thin film scanning calorimetry technique. The films consist of ensembles of self-assembled indium nanostructures whose melting temperatures are strongly influenced by size. We experimentally determine the relationship between a nanostructure's radius and its melting point by combining the measured caloric results with nanostructure size distributions obtained from TEM. The results show a linear melting point depression. Moreover, by looking at the fine structures of these caloric curves, we found the discrete nature of nanostructures during the early stage of thin film growth. The measured heat capacity values show several local maxima at certain temperatures. This suggests that preferred energy states exist among these supported nanostructures on amorphous surfaces. These local maxima are related to each other by increments of one monolayer of indium atoms. These findings could be extended from the magic numbers observed previously in cluster beams studies.