To enhance understanding of the Zintl-Klemm concept, which is useful for characterizing chemical bonding in semimetallic and semiconducting valence compounds, and to more effectively rationalize the structures of Zintl phases, we present a partitioning scheme of the total energy calculated on numerous possible structures of the alkali metal trielides, LiAl, LiTl, NaTl, and KTl, using first-principles quantum mechanical calculations. This assessment of the total energy considers the relative effects of covalent, ionic, and metallic interactions, all of which are important to understand the complete structural behavior of Zintl phases. In particular, valence electron transfer and anisotropic covalent interactions, explicitly employed by the Zintl-Klemm concept, are often in competition with isotropic, volume-dependent metallic and ionic interaction terms. Furthermore, factors including relativistic effects, electronegativity differences, and atomic size ratios between the alkali metal and triel atoms can affect the competition by enhancing or weakening one of the three energetic contributors and thus cause structural variations. This partitioning of the total energy, coupled with analysis of the electronic density of states curves, correctly predicts and rationalizes the structures of LiAl, LiTl, NaTl, and KTl, as well as identifies a pressure-induced phase transition in KTl from its structure, based on [Tl-6](6-) distorted octahedra, to the double diamond NaTl-type.