The Bird and Nanbu systems are particle systems used to approximate the solution of Boltzmann mollified equation. In particular, they have the propagation of chaos property. Following [GM94], we use coupling techniques and resultson branching processes to write an expansion of the error in the propagation of chaos in terms of the number of particles, for slightly more general systems than the ones cited above. As explained in [DMPR] and [DMPR09], this result will lead to the proof of the convergence of U-statistics for these systems.