Thesis advisor: Kathryn Lindsey / We use interval maps to construct pseudo-Anosovs and relate important invariants of each regime. This work builds on techniques of André de Carvalho, Toby Hall, Bill Thurston, and others. We introduce a new perspective on the pseudo-Anosovs created in this way, showing how they constitute the vertices of a tree whose edges encode relations between them. We also characterize the pseudo-Anosovs arising from interval maps, and use this result to reprove a universal lower bound on their stretch factors originally due to Boissy-Lanneau. / Thesis (PhD) — Boston College, 2023. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.