Semiempirical tight binding (TB) and density functional theory (DFT) methods have been used to study the mechanism of single walled carbon nanotube (SWNT) growth. The results are compared with similar calculations on graphene. Both TB and DFT geometry optimized structures of relevance to SWNT growth show that the minimum energy growth mechanism is via the formation of hexagons at the SWNT end. This is similar to the result for graphene where growth occurs via the formation of hexagons at the edge of the graphene flake. However, due to the SWNT curvature, defects such as pentagons are more stable in SWNTs than in graphene. Monte Carlo simulations based on the TB energies show that SWNTs close under conditions that are proper for growth of large defect-free graphene flakes, and that a particle such as a Ni cluster is required to maintain an open SWNT end under these conditions. The calculations also show that the proper combination of growth parameters such as temperature and chemical potential are required to prevent detachment of the SWNTs from the Ni cluster or encapsulation of the cluster by the feedstock carbon atoms.