This article deals with the two-stage stochastic model, which aims at explicitly taking into account uncertainty in optimization problems, that Kong and Schaefer have recently studied for the maximum weight matching problem [N. Kong, A.J. Schaefer, A factor 1/2 approximation algorithm for two-stage stochastic matching problems, European Journal of Operational Research 172(3) (2006) 740–746]. They have proved that the problem is NP-hard, and they have provided a factor View the MathML source12 approximation algorithm. We further study this problem and strengthen the hardness results, slightly improve the approximation ratio and exhibit some polynomial cases. We similarly tackle the maximum weight spanning tree problem in the two-stage setting. Finally, we make numerical experiments on randomly generated instances to compare the quality of several interesting heuristics.