A Brownian motion tree (BMT) model is a Gaussian model whose associated set of covariance matrices is linearly constrained according to common ancestry in a phylogenetic tree. We study the complexity of inferring the maximum likelihood (ML) estimator for a BMT model by computing its ML-degree. Our main result is that the ML-degree of the BMT model on a star tree with $n + 1$ leaves is $2^{n+1}-2n-3$, which was previously conjectured by Am\'endola and Zwiernik. We also prove that the ML-degree of a BMT model is independent of the choice of the root. The proofs rely on the toric geometry of concentration matrices in a BMT model. Toward this end, we produce a combinatorial formula for the determinant of the concentration matrix of a BMT model, which generalizes the Cayley-Pr\"ufer theorem to complete graphs with weights given by a tree.

翻译：布朗运动树（BMT）模型是一种高斯模型，其协方差矩阵集合根据系统发育树中的共同祖先受到线性约束。我们通过计算BMT模型的最大似然（ML）度来研究推断其ML估计量的复杂性。主要结果是：具有$n+1$个叶子的星形树上BMT模型的ML度为$2^{n+1}-2n-3$，这先前由Am\'endola和Zwiernik推测。我们还证明了BMT模型的ML度与根的选择无关。证明依赖于BMT模型中浓度矩阵的环面几何性质。为此，我们给出了BMT模型浓度矩阵行列式的组合公式，该公式将Cayley-Prüfer定理推广到权重由树给定的完全图。