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A numerical simulation utilizing a box_tree code is used to investigate the structure and vertical mode spectrum of finite two-dimensional (2-D) plasma crystals. The overall structural symmetry of the system is examined for various Debye lengths, and a transition from a predominantly hexagonal structure to a structure having concentric rings along the outer edge and hexagonal lattice symmetry in the interior is shown to develop as the Debye length increases. Both the vertical and horizontal oscillation modes for this type of system are investigated, where the horizontal mode spectra are shown to agree with the published results, whereas the vertical mode spectra obtained are shown to agree with an independent analytical method. The fundamental frequency for the vertical modes decreases as the mode number l decreases and is shown to have a maximum corresponding to that which would exist if the system acted in toto as a solid plane. For low-frequency vertical modes, the largest amplitude particle motion is concentrated within a few inner rings, with the outer rings remaining almost motionless. Both of these are in direct contrast with the data obtained for the horizontal modes, where it is shown that the largest amplitude particle motion is concentrated within the inner rings at high frequencies.