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Structural polynomials such as the Tutte polynomial for graphs and the Jones polynomial for knots are essential algebraic objects for studying discrete structures, especially for unlabeled structures or topologies. These polynomial invariants encode structural information and can be represented as vectors or matrices. This allows for analyzing discrete structures with vast data analytic tools. Discrete structures, in particular, trees, emerge in many areas of life science and record important biological information, for example, hierarchical structures in evolution, patterns of cell divisions and differentiation in the development of an organism and the molecular structures of nucleic acids and proteins. In this talk, we introduce an easy-to-compute and interpretable structural polynomial that is a complete invariant for trees. We apply the polynomial to solve problems in molecular evolution and to study nucleic acid structures. We show that this approach detects, for example, distinct evolutionary patterns of seasonal and tropical human influenza virus A H3N2 and the correlation between RNA secondary structures and R-loop formation.