Three ordered structures of the tetragonal BaAl4 type were identified in the Ba-Au-Sn system, from which a unified view of the interplay between the valence electron counts (VECs) and phase stabilities of these three types of derivatives can be developed. The BaNiSn3 (I4mm), ThCr2Si2 (I4/mmm), and CaBe2Ge2 (P4/nmm) type BaAuxSn4-x, phases occurred respectively at x = 0.78(1)-1, 1.38(1)-1.47(1), and 1.52(1)-2.17(1), consistent with theoretical atomic "coloring" analyses that reveal an optimal VEC of similar to 14 for the ThCr2Si2 type but larger and smaller values respectively for the BaNiSn3- and CaBe2Ge2-type structures.