Despite impressive technical and theoretical developments, reconstruction of phylogenetic trees for enormous quantities of molecular data is still a challenging task. A key tool in analyses of large data sets has been the construction of separate trees for subsets (e.g., quartets) of sequences, and subsequent combination of these subtrees into a single tree for the full set (i.e., supertree analysis). Unfortunately, even amalgamating quartets into a supertree remains a computationally daunting task. Assigning weights to quartets to indicate importance or reliability was proposed more than a decade ago, but handling weighted quartets is even more challenging and has scarcely been attempted in the past. In this work we focus on weighted quartet-based approaches. We propose a scheme to assign weights to quartets coming from weighted trees and devise a tree similarity measure for weighted trees based on weighted quartets. We also extend the quartet MaxCut (QMC algorithm) to handle weighted quartets. We evaluate these tools on simulated and real data. Our simulated data analysis highlights the additional information that is conveyed when using the new weighted tree similarity measure, and shows that extending QMC to a weighted setting improves the quality of tree reconstruction. Our analyses of a cyanobacterial data set with wQMC reinforce previous results achieved with other tools.