The growth of narrow single-wall carbon nanotubes through adduction of small carbon clusters is studied using a molecular-dynamics simulation method. Statistical behavior of the growth and defect formation process is analyzed. For C2 dimer colliding onto the side-wall of narrow single-wall nanotubes, it is very easy to get the dimer to be incorporated into the network of the tube during annealing, forming localized topological defects. During long-time annealing at 2300K, thermal fluctuation can cause structural switching among different metastable states and thereby result in energy pulses in the energy vs time curve.