One of the criteria for inferring a species tree from a collection of gene trees, when gene tree incongruence is assumed to be due to incomplete lineage sorting (ILS), is minimize deep coalescence, or MDC. Exact algorithms for inferring the species tree from rooted, binary trees under MDC were recently introduced. Nevertheless, in phylogenetic analyses of biological data sets, estimated gene trees may differ from true gene trees, be incompletely resolved, and not necessarily rooted. In this paper, we propose new MDC formulations for the cases where the gene trees are unrooted/binary, rooted/non-binary, and unrooted/nonbinary. Further, we prove structural theorems that allow us to extend the algorithms for the rooted/binary gene tree case to these cases in a straightforward manner. Finally, we study the performance of these methods in coalescent-based computer simulations
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