The structure of H⁺ (H₂O)₆ is investigated by examining selected low-lying minima with several first principle methods to benchmark the performance of these methods employed in the previous theoretical studies. Interestingly, we found that DFT methods with a moderate basis set follow the trends of MP2 with a large basis set very closely. In additional to the conventional zero point energy estimated with harmonic oscillator approximation, the contribution of vibrational anharmonicity is also investigated via first principle calculations. We found the anharmonicity contribution to the zero point energy varies between 2.7 to 5.0 mhartree and with that two kinds of tree structures (simple and branched tree) are found to be the most stable forms with nearly the same energy. The effects of Ar-attachment on the relative stability of these two tree structures are also examined and we found that the Ar-attached branched tree is more stable than the Ar-attached simple tree by about 1.4 mhartree. We shall also discuss the relevance of our findings with the recent experimental spectra on both bare and Ar-attached species.