We present a systematic study of the temperature and pressure dependence of the growth rate of vertically aligned small diameter (single- and few-walled) carbon nanotube forests grown by thermal chemical vapor deposition over the temperature range 560-800 degrees C and 10(-5) to 14 mbar partial pressure range, using acetylene as the feedstock and Al(2)O(3)-supported Fe nanoparticles as the catalyst. We observe a pressure dependence of P(0.6) and activation energies of <1 eV. We interpret this as a growth rate limited by carbon diffusion in the catalyst, preceded by a pre-equilibrium of acetylene dissociation on the catalyst surface. The carbon nanotube forest growth was recorded by high-resolution real-time optical imaging.