With distributed computing and mobile applications, synchronizing diverging replicas of data structures is a more and more common problem. We use algebraic methods to reason about filesystem operations, and introduce a simplified definition of conflicting updates to filesystems. We also define algorithms for update detection and reconciliation and present rigorous proofs that they not only work as intended, but also cannot be improved on. To achieve this, we introduce a novel, symmetric set of filesystem commands with higher information content, which removes edge cases and increases the predictive powers of our algebraic model. We also present a number of generally useful classes and properties of sequences of commands. While these results are often intuitive, providing exact proofs for them is far from trivial. They contribute to our understanding of this special type of algebraic model, and toward building more complete algebras of filesystem trees and extending algebraic approaches to other data storage protocols. They also form a theoretical basis for specifying and guaranteeing the error-free operation of applications that implement an algebraic approach to synchronization.