The video of a growing fullerene within a carbon nanotube, initiated by a tungsten catalyst, provides a dramatic realization of a complex nanoscale process. While there may be many detailed models which can account for this growth, we propose one of the simplest possible models which is consistent with the major observed features of the growth process. In particular, we assume that the fullerene is immersed in a carbon vapor environment, and that the growth occurs as a consequence of the diffusion of the carbon vapor into the fullerene. Moreover, we assume that the classical diffusion equation applies in the region exterior to the fullerene and that a standard Stefan condition applies at the moving fullerene surface. We assume that the gaseous medium through which the carbon atoms diffuse is represented through the value of the diffusion coefficient D appearing in the classical diffusion equation. We also assume that the influence of the catalyst is felt through the value of the constant appearing in the Stefan condition. Based on these assumptions, we derive simple similarity solutions for both spherical and ellipsoidal fullerenes which are entirely consistent with the observations. A corresponding analysis is provided for the longitudinal growth of a carbon nanotube.