When to use

A tree zipper is a cursor for immutable trees. State = (focus, trail).

Rules

focus is the current subtree
trail is a list of "breadcrumbs" describing the path from root to focus
Each crumb remembers the parent's value plus the siblings NOT taken
down_left/down_right push a crumb and make the chosen child the new focus
up pops the top crumb, rebuilds the parent, and makes that parent the new focus
set_value replaces the focused subtree's value
to_tree walks all the way up to rebuild the whole tree
Key invariant: you can ALWAYS reconstruct the full original tree from (focus, trail)
Equality of two zippers = equality of the fully-reconstructed trees, NOT of the raw (focus, trail) pairs
NEVER compare raw (focus, trail) pairs for equality