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The MLE of a variable (parameter) is the value that maximizes the likelihood function. The approach of Felsenstein (1981) is to estimate the parameters of the substitution process and the branch lengths for each tree. Then choose among tree topologies by considering their relative likelihoods. The tree topology with the greatest likelihood is chosen as the MLE of the phylogeny. These might be gamma distributions. See also http://mw511.biol.berkeley.edu/like/mle/