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Phylogenetic trees are commonly reconstructed based on hard optimization problems such as maximum parsimony (MP) and maximum likelihood (ML). Conventional MP heuristics for producing phylogenetic trees produce good solutions within reasonable time on small datasets (up to a few thousand sequences), while ML heuristics are limited to smaller datasets (up to a few hundred sequences). However, since MP (and presumably ML) is NP-hard, such approaches do not scale when applied to large datasets. In this paper, we present a new technique called Recursive-Iterative-DCM3 (Rec-I-DCM3), which belongs to our family of disk-covering methods (DCMs). We tested this new technique on ten large biological datasets ranging from 1,322 to 13,921 sequences and obtained dramatic speedups as well as significant improvements in accuracy (better than 99.99%) in comparison to existing approaches. Thus, high-quality reconstructions can be obtained for datasets at least ten times larger than was previously possible.