Controlling the structure of single-wall carbon nanotubes during their synthesis by chemical vapor deposition remains a challenging issue. Here, using a specific synthesis protocol and ex situ transmission electron microscopy, we perform a statistical analysis of the structure of the tubes and of the catalyst particles from which they grow. We discriminate two nucleation modes, corresponding to different nanotube-particle junctions, that occur independently of the particle size. With the support of tight binding calculations, we show that a direct control of the nanotube diameter by the particle can only be achieved under growth conditions close to thermodynamic equilibrium.