Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Functional programming, design and analysis of algorithms / 3 / d / iii

A list unfold needs an explicit state describing the unvisited work. Use a list of pending rose trees, with the next tree first. Emit that tree's root label, and replace it by its children followed by the remaining pending trees:

dftU :: RTree a -> [a]
dftU t = unfold null label advance [t]
  where
    label :: [RTree a] -> a
    label (Node x _ : _) = x

    advance :: [RTree a] -> [RTree a]
    advance (Node _ children : rest) = children ++ rest

The boxed state transition is $\boxed{\mathrm{Node}(x,children):rest⟼children++rest}$, with output $x$. The stopping test prevents either helper from being applied to [].

The loop invariant is that concatenating the depth-first traversals of the pending trees gives exactly the unproduced output. Removing the next root emits its first label; replacing that tree by its children preserves the remainder of that preorder traversal. Adding children at the front realizes an explicit-stack depth-first tree traversal; placing them at the end would give a different visiting order. The first states for the source example have roots [1], [2,5,6], [3,4,5,6], [4,5,6], [5,6], [6], [7], [].