Multi-zone chemical evolution models (CEMs), differing in the nucleosynthesis prescriptions (yields) and prescriptions of star formation, have been computed for the Milky Way. All models fit the observed O/H and Fe/H gradients well and reproduce the main characteristics of the gas distribution, but they are also designed to do so. For the C/H gradient the results are inconclusive with regards to yields and star formation. The C/Fe and O/Fe vs. Fe/H, as well as C/O vs. O/H trends predicted by the models for the solar neighbourhood zone were compared with stellar abundances from the literature. For O/Fe vs. Fe/H all models fit the data, but for C/O vs. O/H, only models with increased carbon yields for zero-metallicity stars or an evolving initial mass function provide good fits. Furthermore, a steep star formation threshold in the disc can be ruled out since it predicts a steep fall-off in all abundance gradients beyond a certain galactocentric distance (∼13 kpc) and cannot explain the possible flattening of the C/H and Fe/H gradients in the outer disc seen in observations. Since in the best-fit models the enrichment scenario is such that carbon is primarily produced in low-mass stars, it is suggested that in every environment where the peak of star formation happened a few Gyr back in time, winds of carbon-stars are responsible for most of the carbon enrichment. However, a significant contribution by zero-metallicity stars, especially at very early stages, and by winds of high-mass stars, which are increasing in strength with metallicity, cannot be ruled out by the CEMs presented here. In the solar neighbourhood, as much as 80%, or as little as 40% of the carbon may have been injected to the interstellar medium by low-and intermediate-mass stars. The stellar origin of carbon remains an open question, although production in low-and intermediate-mass stars appears to be the simplest explanation of observed carbon abundance trends.