Reconstructing the shape of a tree from observed dissimilarity data
Advances in Applied Mathematics
A few logs suffice to build (almost) all trees (l): part I
Random Structures & Algorithms
Information Theory and Reliable Communication
Information Theory and Reliable Communication
Provably Fast and Accurate Recovery of Evolutionary Trees through Harmonic Greedy Triplets
SIAM Journal on Computing
Maximum likelihood on four taxa phylogenetic trees: analytic solutions
RECOMB '03 Proceedings of the seventh annual international conference on Research in computational molecular biology
New results on optimizing rooted triplets consistency
Discrete Applied Mathematics
The complexity of inferring a minimally resolved phylogenetic supertree
WABI'10 Proceedings of the 10th international conference on Algorithms in bioinformatics
Computing a Smallest Multilabeled Phylogenetic Tree from Rooted Triplets
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
The Complexity of Inferring A Minimally Resolved Phylogenetic Supertree
SIAM Journal on Computing
Kernel and fast algorithm for dense triplet inconsistency
Theoretical Computer Science
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We consider the problem of finding the maximum likelihood rooted tree of three species under a molecular clock symmetric model of substitution of 2-state characters. For identically distributed rates per site this is probably the simplest phylogenetic estimation problem, and it is readily solved numerically. Analytic solutions, on the other hand, were obtained only recently by Yang [Complexity of the simplest phylogenetic estimation problem, Proc. Roy Soc. London Ser. B 267 (2000) 109-119]. In this work we provide analytic solutions for any distribution of rates across sites, provided the moment generating function of the distribution is strictly increasing over the negative real numbers. This class of distributions includes, among others, identical rates across sites, as well as the Gamma, the uniform, and the inverse Gaussian distributions. Our work therefore generalizes Yang's solution and our derivation of the analytic solution is substantially simpler. We use the Hadamard conjugation to prove a general statement about the edge lengths of any neighboring pair of leaves in any phylogenetic tree (on three or more taxa). We then employ this relation, in conjunction with the convexity of an entropy-like function, to derive the analytic solution.