  We study the problem of maximum likelihood estimation given one data sample
($n=1$) over Brownian Motion Tree Models (BMTMs), a class of Gaussian models on
trees. BMTMs are often used as a null model in phylogenetics, where the
one-sample regime is common. Specifically, we show that, almost surely, the
one-sample BMTM maximum likelihood estimator (MLE) exists, is unique, and
corresponds to a fully observed tree. Moreover, we provide a polynomial time
algorithm for its exact computation. We also consider the MLE over all possible
BMTM tree structures in the one-sample case and show that it exists almost
surely, that it coincides with the MLE over diagonally dominant M-matrices, and
that it admits a unique closed-form solution that corresponds to a path graph.
Finally, we explore statistical properties of the one-sample BMTM MLE through
numerical experiments.
