Rose tree

A rose tree is an ordered rooted tree in which each vertex can have any finite number of children in a rooted tree. A labelled Haskell representation is data RTree a = Node a [RTree a]. Its tree fold has type (a -> [b] -> b) -> RTree a -> b: recursively fold the children, then combine their results with the current label. Unlike a binary tree, a rose tree has no fixed pair of child positions.

A rose tree is perfect when all its leaves of a rooted tree have the same depth. This permits unequal branching counts and unary branches; perfectness alone does not require every internal vertex to have the same number of children. A bottom-up certificate returns Just 0 at a leaf, Just (h+1) when all child certificates are Just h, and Nothing otherwise. Structural induction proves that a certificate exists exactly when the tree is perfect: all root-to-leaf routes acquire one edge when attached to the new root.