2-Adic Tree Embeddings: A Modeling and Optimization Framework for Computational Phylogenetics

Phylogenetic inference is a fundamental problem in computational biology, with applications ranging from epidemiology and oncology to developmental biology and comparative genomics. Many major phylogenetic problems are NP-hard; consequently, existing methods often explore tree space using local search or MCMC based on a limited repertoire of combinatorial rearrangements. However, the size and ruggedness of tree space make it difficult for such heuristics to reliably reach optimal or near-optimal solutions. Recently, there has been growing interest in alternative approaches that embed trees into numerical spaces and apply numerical optimization techniques. However, such methods are still limited in practical applicability. We propose a new approach to phylogenetic inference based on embeddings of trees into the non-Archimedean space of 2-adic numbers. The hierarchical organization of 2-adic space naturally encodes tree topologies, while its rich topological, algebraic, and measure-theoretic structure provides a natural foundation for a coherent theory of tree calculus and optimization. Here, we introduce 2-adic tree embedding, establish their basic topological, probabilistic, and algebraic properties, and show that they support natural representations of tree distributions, algebraic operations, and broad classes of tree rearrangements extending standard operations such as NNI, SPR, and TBR. Building on these results, we introduce a general local search framework in which tree neighborhoods are generated by parametric families of 2-adic polynomials. We show that even infinitely many transformations induce only quadratically many distinct neighboring trees, which can be generated in polynomial time. These results suggest a new mathematically rigorous and flexible approach to representing and exploring tree space.