A detailed theoretical study of carbon nanotube (NT) forest growth by chemical vapor deposition is given, including (i) ballistic mode of carbon species impingement into the NT surface, (ii) the carbon diffusion over NT surface and through the metal nanoparticle, and (iii) the temperature drop at the NT tip occurring with increase in NT length. For typical NT forest growth parameters the ballistic flux of carbon species impinging into the NT surface decays quasiexponentially within several microns from the top. A variety of feasible growth modes, ranging from linear to exponential versus time, is predicted agreeing well with reported experiments. The presence of a metal nanoparticle is shown to shift NT growth from being surface diffusion controlled to being controlled by bulk diffusion through the nanoparticle. For typical growth conditions the growth rate is shown to be controlled simultaneously by surface diffusion over NT surface and bulk diffusion of carbon through metal nanoparticle. However, even in specific cases where NT growth rate is controlled by bulk diffusion through the nanoparticle the initial stage may be controlled by surface diffusion, as revealed by the exponential change in NT length with time. A parametric study of the growth rate of NT forest with metal nanoparticles held at the NT tips as a function of temperature reveals the existence of a maximum near 1050-1100 K, agreeing with reported experimental data. A thermal analysis based upon the heat conductance equation shows that with NT forest growth the temperature of the NT tips decreases, leading to growth deceleration and termination. Our study shows that the larger the pressure the smaller the NT forest height that may be grown. In particular, for pressures approximate to 10(5) Pa the NT tips should be "frozen" even at a length of a few microns, disabling further NT growth. In contrast, under low pressures of approximate to 10(3) Pa NT forest of several dozens of microns may be successfully grown without significant growth deceleration. (C) 2003 American Institute of Physics.