We first consider the reconstruction problem for a ferromagnetic Ising model on a finite binary tree: infer an internal spin given only the spins at the leaves. The maximum a posteriori (MAP) estimator is canonical, but depends on unknown edge couplings. We show that in a suitable low-noise regime the estimator is insensitive to parameter misspecification, up to a controlled error. Next, we leverage this robustness to analyze the optimization landscape of the maximum likelihood estimator (MLE) for the edge parameters. While the population log-likelihood function is generally non-concave, we prove that it becomes locally strongly concave around the true parameters in this regime. This provides a theoretical justification for the efficacy of gradient-based optimization methods widely used in phylogenetic inference. Joint work with David Clancy, Hanbaek Lyu, and Allan Sly.