Supertree methods are used to construct a large tree over a large set of taxa, from a set of small trees over overlapping subsets of the complete taxa set. Since accurate reconstruction methods are currently limited to a maximum of few dozens of taxa, the use of a supertree method in order to construct the tree of life is inevitable. Perhaps the simplest version of this that is widely applicable, yet quite challenging, is quartet based reconstruction. This problem lies at the root of many tree reconstruction methods and theoretical as well as experimental results have been reported. Nevertheless, dealing with false, conflicting quartets remains problematic. In this paper, we describe an algorithm for constructing a tree from a set of input quartet trees even with a significant fraction of errors. Our algorithm is based on a divide and conquer algorithm where our divide step uses a semi- definite formulation of max cut.We remark that this builds on previous work of ours [28] for piecing together trees from rooted triples. The recursion for quartets, however, is more complicated in that even with completely consistent quartets the problem is NP-hard.This complexity leads to several issues and some solutions of possible independent interest.