We present an extensive investigation of the epitaxial growth of Au-assisted axial heterostructure nanowires composed of group IV and III-V materials and derive a model to explain the overall morphology of such wires. By analogy with 2D epitaxial growth, this model relates the wire morphology (i.e., whether it is kinked or straight) to the relationship of the interface energies between the two materials and the particle. This model suggests that, for any pair of materials, it should be easier to form a straight wire with one interface direction than the other, and we demonstrate this for the material combinations presented here. However, such factors as kinetics and the use of surfactants may permit the growth of straight double heterostructure nanowires. Finally, we demonstrate that branched nanowire heterostructures, also known as nanotrees, can be successfully explained by the same model.