Piecing Together the Tree of Life — with Tropical Geometry

by Dr. Andrei Comăneci (Winner of YRA 2025)

How do we find consensus when multiple evolutionary trees tell different stories? In computational biology, phylogenetic trees often conflict due to noise, differing data, or model assumptions. Classical methods like majority-rule consensus work by retaining only those branches shared by most trees—but they ignore branch lengths and often break down in the presence of “rogue taxa,” species that jump around in the tree.

My research introduces a novel approach based on tropical geometry, a field where “addition” is defined as taking the maximum, and “multiplication” becomes addition. This sounds strange, but it enables a new geometric structure—tropical convexity—that is surprisingly well-suited to modeling phylogenetic trees.

Instead of “voting” on branches, we treat trees as points in a tropical space and ask: what tree lies closest to all input trees under a tropical distance? This reframes consensus as a location problem. I proved that for a broad class of tropical distances, the optimal consensus always lies in the tropical convex hull of the inputs—preserving shared structure, while reducing sensitivity to rogue taxa.

To achieve this, we introduced a new class of functions called tropically quasiconvex. These yield dissimilarity measures that don’t necessarily satisfy the triangle inequality, yet are structured enough to reflect the tropical geometry of tree space. This makes them well-suited to applications where classical distances fall short. The framework also extends naturally to situations with missing data, by treating each input not as a single point but as a set—turning the problem into one of finding a location relative to multiple sets. What makes this particularly elegant is that key combinatorial features of phylogenetic trees—such as shared substructures—correspond to tropically convex sets.

The result: a robust, efficient framework for consensus tree construction that naturally handles edge lengths, avoids manual constraints, and scales to large datasets. It’s a fresh example of how abstract mathematics—tropical geometry—can tackle real-world biological and optimization problems.