Abstract
How do phylogenetic reconstruction algorithms go astray when they return incorrect trees? This simple question has not been answered in detail, even for maximum parsimony (MP), the simplest phylogenetic criterion. Understanding MP has recently gained relevance in the regime of extremely dense sampling, where each virus sample commonly differs by zero or one mutation from another previously sampled virus. Although recent research shows that evolutionary histories in this regime are close to being maximally parsimonious, the structure of their deviations from MP is not yet understood. In this paper, we develop algorithms to understand how the correct tree deviates from being MP in the densely sampled case. By applying these algorithms to simulations that realistically mimic the evolution of SARS-CoV-2, we find that simulated trees frequently only deviate from maximally parsimonious trees locally, through simple structures consisting of the same mutation appearing independently on sister branches. We leverage this insight to design approaches for sampling near-MP trees and using them to efficiently estimate clade supports.