We propose a novel method for the synthesis of computationally efficient multi-stage model predictive controllers based on scenario trees for uncertain linear systems. As for several uncertainties and long prediction horizons the scenario tree can become very large, we consider a situation where a robust horizon is used, i.e., the scenario tree branches only over the first stages and then a nominal model is assumed. We provide an analysis and a simple but efficient algorithm to ensure recursive feasibility in this situation by introducing constraint tightening only after the robust horizon. The resulting control law is non-parametrized which increases the size of the feasible set of the resulting multi-stage controller. The online computational complexity is reduced significantly with respect to other multi-stage based controllers.