For the Ga-assisted growth of GaAs nanowires on Si(111) substrates by molecular beam epitaxy, growth temperature, As flux, and Ga flux have been systematically varied across the entire window of growth conditions that result in the formation of nanowires. A range of GaAs structures was observed, progressing from pure Ga droplets under negligible As flux through horizontal nanowires, tilted nanowires, vertical nanowires, and nanowires without droplets to crystallites as the As flux was increased. Quantitative analysis of the resulting sample morphology was performed in terms of nanowire number and volume density, number yield and volume yield of vertical nanowires, diameter, length, as well as the number and volume density of parasitic growth. The result is a growth map that comprehensively describes all nanowire and parasitic growth morphologies and hence enables growth of nanowire samples in a predictive manner. Further analysis indicates the combination of global Ga flux and growth temperature determines the total density of all objects, whereas the global As/Ga flux ratio independently determines the resultant sample morphology. Several dependencies observed here imply that all objects present on the substrate surface, i.e. both nanowires and parasitic structures, originate from Ga droplets.