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The voting patterns of the nine justices on the United States Supreme Court continue to fascinate and perplex observers of the Court. While it is commonly understood that the division of the justices into a liberal branch and a conservative branch inevitably drives many case outcomes, there are finer, less transparent divisions within these two main branches that have proven difficult to extract empirically. This study imports methods from evolutionary biology to help illuminate the intricate and often overlooked branching structure of the justices’ voting behavior. Specifically, phylogenetic tree estimation based on voting disagreement rates is used to extend ideal point estimation to the non-Euclidean setting of hyperbolic metrics. After introducing this framework, comparing it to one- and two-dimensional multidimensional scaling, and arguing that it flexibly captures important higher-dimensional voting behavior, a handful of potential ways to apply this tool are presented. The emphasis throughout is on interpreting these judicial trees and extracting qualitative insights from them. 

In 2004, Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map with a weighted variant, we show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter and Speyer envisioned. Moreover, we provide a geometric interpretation in terms of configurations of points on rational normal curves and construct a finite tropical basis that yields an explicit characterization of weighted dissimilarity vectors.