Abstract: Let T_n be a set of trees with n vertices. The increasing orders in terms of the minimal energies of the trees in T_n had been considered and two conjectures were proposed. The first one is the comparison of the energies of two graphs in T_n. The other is about the trees with the 10th to 12th minimal energies for n≥7 117 599. This paper proves that the two conjectures are true, and obtain the first 12 trees within T_n for n≥7 117 599.

Key words: minimal energy, ordering, tree