Phylogenetic trees represent evolutionary histories and have many important applications in biology, anthropology and criminology. The branching structure of the tree encodes the order of evolutionary divergence, and the branch lengths denote the time between divergence events. The target of interest in phylogenetic tree inference is high-dimensional, but the real challenge is that both the discrete (tree topology) and continuous (branch lengths) components need to be estimated. While decomposing inference on the topology and branch lengths has been historically popular, the mathematical and algorithmic developments of the last 15 years have provided a new framework for holistically treating uncertainty in tree inference. I will discuss how we can leverage these developments to construct a confidence set for the Fréchet mean of a distribution with support on the space of phylogenetic trees. The sets have good coverage and are efficient to compute. I will conclude by applying the procedure to revisit an HIV forensics investigation, and to assess our confidence in the geographical origins of the Zika virus.