**Summary:** Joseph Felsenstein discusses the application of maximum likelihood methods for estimating evolutionary trees from DNA sequences. He presents a computationally feasible method for finding maximum likelihood estimates and describes a computer program that implements this method. This approach is more reliable than parsimony methods, which can give misleading results when evolutionary rates vary among lineages. Maximum likelihood methods also allow testing hypotheses about the constancy of evolutionary rates and provide an estimate of the error in the tree. The paper introduces a probabilistic model of DNA evolution and develops an algorithm for computing the likelihood of a given tree. This algorithm is computationally feasible for DNA sequences, unlike protein sequences. Felsenstein proposes an iterative method for altering the tree to increase likelihood, forming the basis of a computer program for maximum likelihood estimation of evolutionary trees from DNA or RNA sequences. The paper also discusses the challenges of modeling DNA evolution, including the difficulty of computing probabilities for complex evolutionary trees. It emphasizes the importance of assuming independence of changes at different sites in the sequence, which simplifies computation. The probability of a given set of sequences on a tree is calculated site by site, with the overall probability being the product of these site probabilities. The paper presents a specific example of a tree and describes how to compute the likelihood of the data given the tree. The method assumes that after speciation, lineages evolve independently, and the same stochastic process of base substitution applies in all lineages. The paper concludes that maximum likelihood methods provide a more accurate and statistically sound approach to estimating evolutionary trees compared to parsimony methods.**Summary:** Joseph Felsenstein discusses the application of maximum likelihood methods for estimating evolutionary trees from DNA sequences. He presents a computationally feasible method for finding maximum likelihood estimates and describes a computer program that implements this method. This approach is more reliable than parsimony methods, which can give misleading results when evolutionary rates vary among lineages. Maximum likelihood methods also allow testing hypotheses about the constancy of evolutionary rates and provide an estimate of the error in the tree. The paper introduces a probabilistic model of DNA evolution and develops an algorithm for computing the likelihood of a given tree. This algorithm is computationally feasible for DNA sequences, unlike protein sequences. Felsenstein proposes an iterative method for altering the tree to increase likelihood, forming the basis of a computer program for maximum likelihood estimation of evolutionary trees from DNA or RNA sequences. The paper also discusses the challenges of modeling DNA evolution, including the difficulty of computing probabilities for complex evolutionary trees. It emphasizes the importance of assuming independence of changes at different sites in the sequence, which simplifies computation. The probability of a given set of sequences on a tree is calculated site by site, with the overall probability being the product of these site probabilities. The paper presents a specific example of a tree and describes how to compute the likelihood of the data given the tree. The method assumes that after speciation, lineages evolve independently, and the same stochastic process of base substitution applies in all lineages. The paper concludes that maximum likelihood methods provide a more accurate and statistically sound approach to estimating evolutionary trees compared to parsimony methods.