Density functional theory (DFT) calculations of the nucleation of (5, 5) and (10, 0) single-walled carbon nanotubes (SWNTs) on a 55 atom nickel cluster (Ni 55 ) showed that it requires a larger chemical potential to grow a carbon island (which is the precursor to the SWNTs) on the cluster than to extend the island into a SWNT or to have the carbon atoms dispersed on the cluster surface. Hence, in the thermodynamic limit the island will only form once the (surface of the) cluster is saturated with carbon, and the island will spontaneously form a SWNT at the chemical potentials required to create the island. The DFT (zero Kelvin) and tight binding Monte Carlo (1000 K) also show that there is a minimum cluster size required to support SWNT growth, and that this cluster size can be used to control the diameter, but probably not the chirality, of the SWNT at temperatures relevant to carbon nanotube growth. It also imposes a minimum size of clusters that are used in SWNT regrowth.