Explore the tropical geometry of phylogenetic tree spaces in this 59-minute lecture by Anthea Monod. Delve into the BHV (Billera Holmes Vogtmann) space, a moduli space of phylogenetic trees with applications in computational biology, computer vision, combinatorics, and category theory. Discover the homeomorphism between BHV space and the Grassmannian using tropical geometry, and learn how this tropical structure enhances computational studies. Examine the coincidence between BHV space and the tropical Grassmannian, and gain insights into recent research on the tropical Grassmannian as a metric space. Understand the practical implications of these findings on probabilistic and statistical studies of phylogenetic tree sets. Cover topics such as notation, terminology, phylogenetic tree origins, tree comparison methods, BHV space construction, the BHV metric, tropical geometry examples, statistical challenges, and the tropical metric's stability.