Ordering of the trees by minimal energies

The ordering of the trees with n vertices according to their minimal energies is investigated by means of a quasi-ordering relation and the theorem of zero points. We deduce the first 9 trees for a general case with n ≥ 46. We obtain the first 12, 11, n + 6, 17, 15, and 12 trees for 7117598 ≥ n ≥ 26, 25 ≥ n ≥ 18, 17 ≥ n ≥ 11, n = 10, n = 9, and n = 8, respectively. For n = 7, we list all the trees in the increasing order of their energies. The maximal diameters of the trees with minimal energies obtained here are 4 for n ≥ 18 and 5 for 17 ≥ n ≥ 8, respectively. For the trees under consideration, the ones with smaller diameters have smaller energies. In addition, we in part prove a conjecture proposed by Zhou and Li (J. Math. Chem. 39:465–473, 2006).