This paper presents a formal construction of dynamic merkle-trees and a deep-dive into theirmathematical structure. In doing so, new and interesting artefacts are presented as well as novel security proof constructions that enable proofs for a full range of tree transformations including append, update and deletion of leaf nodes (without requiring knowledge of those nodes). Novel concepts are explored including "perfect trees", "sub-trees", "sub-roots" and "flat coordinates" through various lemmas, theorems and algorithms. In particular, a "flat-tree" implementation of merkle-trees is presented suitable for storing trees as a contiguous block of memory that can grow and shrink from the right-side. Of note, a "long-tree" implementation is presented which permits arbitrarily large tree construction in logarithmic space and time complexity using a novel algorithm. Finally, a reference implementation accompanies this paper which contains a fully implemented and thoroughly tested implementation of dynamic merkle-trees.