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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 ontrees. BMTMs are often used as a null model in phylogenetics, where theone-sample regime is common. Specifically, we show that, almost surely, theone-sample BMTM maximum likelihood estimator (MLE) exists, is unique, andcorresponds to a fully observed tree. Moreover, we provide a polynomial timealgorithm for its exact computation. We also consider the MLE over all possibleBMTM tree structures in the one-sample case and show that it exists almostsurely, that it coincides with the MLE over diagonally dominant M-matrices, andthat it admits a unique closed-form solution that corresponds to a path graph.Finally, we explore statistical properties of the one-sample BMTM MLE throughnumerical experiments.

Maximum Likelihood Estimation for Brownian Motion Tree Models Based on One Sample