  Reconstructing evolutionary trees from molecular sequence data is a
fundamental problem in computational biology. Stochastic models of sequence
evolution are closely related to spin systems that have been extensively
studied in statistical physics and that connection has led to important
insights on the theoretical properties of phylogenetic reconstruction
algorithms as well as the development of new inference methods. Here, we study
maximum likelihood, a classical statistical technique which is perhaps the most
widely used in phylogenetic practice because of its superior empirical
accuracy.
  At the theoretical level, except for its consistency, that is, the guarantee
of eventual correct reconstruction as the size of the input data grows, much
remains to be understood about the statistical properties of maximum likelihood
in this context. In particular, the best bounds on the sample complexity or
sequence-length requirement of maximum likelihood, that is, the amount of data
required for correct reconstruction, are exponential in the number, $n$, of
tips---far from known lower bounds based on information-theoretic arguments.
Here we close the gap by proving a new upper bound on the sequence-length
requirement of maximum likelihood that matches up to constants the known lower
bound for some standard models of evolution.
  More specifically, for the $r$-state symmetric model of sequence evolution on
a binary phylogeny with bounded edge lengths, we show that the sequence-length
requirement behaves logarithmically in $n$ when the expected amount of mutation
per edge is below what is known as the Kesten-Stigum threshold. In general, the
sequence-length requirement is polynomial in $n$. Our results imply moreover
that the maximum likelihood estimator can be computed efficiently on randomly
generated data provided sequences are as above.
