Various embodiments of the present invention are directed to an instant snapshot-tree-copy operation that, when directed to an existing snapshot tree, generates a data-structure framework for a copy of the existing snapshot tree that can be immediately accessed while a background-copy process copies data from the existing snapshot tree to the copy snapshot tree. The background-copy process copies data from the existing snapshot tree to the copy snapshot tree in top-down fashion, from the source logical device through each of the snapshot logical devices. While copying each logical device, the background-copy process sequentially copies data-storage units from a corresponding logical device of the existing snapshot tree to the copy-snapshot-tree logical device, making sure to adjust chains of dependencies within the copy snapshot tree to reflect modifications to the snapshot copy tree that occur independently from the background-copy process. A host computer may immediately access the copy snapshot tree, both for READ and WRITE access, even before completion of the background-copy process.