Menu Tree Changes

A restaurant menu is a tree. Each node is a dictionary {"key": str, "value": int, "active": bool, "children": [...]}: the root is the whole menu, inner nodes are categories, and leaves are items. A whole menu may also be None (no menu at all).

Given the old_menu and the new_menu, return the number of nodes that changed. Nodes are matched like this:

• The two roots match if they have the same key.
• Two children match if their parents match and they have the same key. The order of children does not matter, and sibling keys are unique.

Then count:

• every matched pair whose value or active differs, counted once per pair;
• every node in the old menu with no match (it was removed), including all of its descendants;
• every node in the new menu with no match (it was added), including all of its descendants.

A node whose own fields changed does not make its children count as changed; they are still compared individually.