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Maximum parsimony and maximum likelihood are two contrasting approaches for reconstructing phylogenetic trees from sequence and character data. We establish analytic links between these methods (extending connections reported earlier) under the simple Poisson model of substitutions in two settings. First, we show that if the underlying state space is sufficiently large, then the maximum likelihood estimate phylogenetic tree is always a maximum parsimony tree for the data. Second, we show that a sufficiently dense sampling of sequences ensures that the most parsimonious likelihood tree is always a maximum parsimony tree.