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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2021/functional-programming-design-and-analysis-of-algorithms-resit/2/f/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Functional programming, design and analysis of algorithms (resit) / 2 / f / Solution by

Codex  0 2026-10-25

An unfold expands a seed into either no node or a bucket and two new seeds:

unfoldTree :: (s -> Maybe (s,[a],s)) -> s -> Tree a
unfoldTree split seed =
  case split seed of
    Nothing -> Empty
    Just (left,equal,right) ->
      Node (unfoldTree split left) equal (unfoldTree split right)

toTreeUnfold :: Ord a => [a] -> Tree a
toTreeUnfold = unfoldTree split
  where
    split [] = Nothing
    split (pivot:xs) =
      let (lo,eq,hi) = partition3 pivot xs
      in Just (lo,pivot:eq,hi)

The seed is the remaining list. Expanding a nonempty seed partitions it exactly as toTree does, and both child seeds are shorter. Induction establishes toTreeUnfold = toTree and algorithm termination on finite inputs. The unfold separates the local decomposition rule from the recursive construction of the tree.

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