The collapse of Lennard-Jones homopolymers is investigated by means of Monte Carlo simulations and the inherent structure/superposition approach, with special emphasis on finite size effects. At thermal equilibrium, the polymers undergo a series of phase changes from the zero temperature folded state to a coexistence state, a molten globule state, the coil state, and finally to a high-temperature "soft" state where the bond lengths vary significantly from their equilibrium value. The correlation between the thermodynamic characteristics of the polymers and those of clusters is interpreted in terms of the energy landscapes of the two systems.