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Joined 2026-09-21 Message
Incoming links: Tree fold
Difference list 2026-10-24
Represent a list by the function
\ys -> xs ++ ys. Concatenation becomes function composition, the empty list is represented by id, and applying to [] recovers the list. Only functions of this tail-prepending form are valid representations; arbitrary functions of the displayed type need not satisfy the invariant. A higher-order tree fold using difference lists produces an inorder traversal in linear time on every tree shape. Leaf-labelled binary tree 2026-10-24
A binary tree with each data label stored at a leaf of a tree. One useful Haskell datatype has constructors
One a for a leaf and Two (Tree a) (Tree a) for an internal node. Every finite inhabitant has at least one leaf, so this datatype cannot encode an empty leaf sequence. Its natural tree fold has a leaf operation a -> b and an internal operation b -> b -> b; replacing the two constructors by those operations defines the structural recursion.For the paper's leaf-labelled binary tree, the combining operation receives the results of the two subtrees, and the other argument handles a leaf:
foldTree :: (a -> b) -> (b -> b -> b) -> Tree a -> b
foldTree f g (One x) = f x
foldTree f g (Two l r) = g (foldTree f g l) (foldTree f g r)Its constructor equation is
foldTree One Two = id. More generally, a function h satisfying h (One x) = f x and h (Two l r) = g (h l) (h r) equals foldTree f g on finite trees, by structural induction. This is the tree fold specialized to a datatype with labels only at leaves, rather than at internal nodes.Both versions preserve left-to-right leaf order:
flatten :: Tree a -> [a]
flatten (One x) = [x]
flatten (Two l r) = flatten l ++ flatten r
flattenFold :: Tree a -> [a]
flattenFold = foldTree (\x -> [x]) (++)The tree fold replaces each leaf by a singleton list and each internal node by list concatenation. The defining equations of
flattenFold are consequently those of flatten, proving their equality by structural induction.A tree fold replaces each constructor by its interpretation:Induction on the finite tree shows equality with
foldTree :: (b -> [a] -> b -> b) -> b -> Tree a -> b
foldTree combine zero Empty = zero
foldTree combine zero (Node left equal right) =
combine (foldTree combine zero left)
equal
(foldTree combine zero right)
fromTreeFold :: Tree a -> [a]
fromTreeFold = foldTree (\left equal right -> left ++ equal ++ right) []fromTree: the empty tree produces the empty output, and the combining function concatenates the two correctly flattened children around the node's bucket. The result type is independent of the key type; other choices can compute a size or another summary using the same structural recursion.Use the source's constructor order: left subtree, root value, right subtree.A tree fold replaces the two constructors by an algebra on already folded children. A tree unfold expands a seed either into the empty tree or into a root value and two new seeds. The definitions themselves work on arbitrary binary trees; preserving the binary search tree ordering requires the supplied algebra or seed expansion to respect its bounds.
data Tree a = Tree (Tree a) a (Tree a) | Empty
foldTree :: (b -> a -> b -> b) -> b -> Tree a -> b
foldTree node empty Empty = empty
foldTree node empty (Tree l x r) =
node (foldTree node empty l) x (foldTree node empty r)
unfoldTree :: (s -> Maybe (s,a,s)) -> s -> Tree a
unfoldTree next seed = case next seed of
Nothing -> Empty
Just (ls,x,rs) ->
Tree (unfoldTree next ls) x (unfoldTree next rs)The direct tree fold for an inorder traversal isBy structural induction, the child lists are ascending; the binary search tree property places every left value below the root and every right value above it. The displayed list concatenation therefore gives an ascending list containing every stored value exactly once.
flattenFold :: Tree a -> [a]
flattenFold = foldTree (\ls x rs -> ls ++ (x:rs)) []For a version with linear time complexity on every tree shape, use an accumulator:This also has a higher-order tree fold expression using difference lists:Its folded value is a function that prepends the inorder traversal to a supplied tail. The accumulator invariant is
flatten :: Tree a -> [a]
flatten t = onto t []
where
onto Empty zs = zs
onto (Tree l x r) zs = onto l (x : onto r zs)flattenLinear :: Tree a -> [a]
flattenLinear t =
foldTree (\l x r -> l . (x:) . r) id t []onto t zs = flattenFold t ++ zs. Each node contributes one constructor, giving time to consume the output. By contrast, the direct list-valued fold can take on a left-skewed tree because ++ repeatedly copies left traversals.A tree fold can compute insertion while retaining the unchanged children by carrying a pair. The following fold-derived implementation preserves the lazy evaluation behaviour of part (f):The invariant of the folded pair is
insertViaFold :: Ord a => a -> Tree a -> Tree a
insertViaFold q t = snd (foldTree node base t)
where
base = (Empty, Tree Empty q Empty)
node ~(l,li) x ~(r,ri) = (original,updated)
where
original = Tree l x r
updated = case compare q x of
LT -> Tree li x r
EQ -> original
GT -> Tree l x ri(original subtree, subtree with q inserted). The empty case establishes it. At a node, the first components reconstruct the original tree, and the second components supply whichever child insertion is needed. The irrefutable pair patterns ~(l,li) and ~(r,ri) avoid demanding an unused child's fold. Induction on finite approximations therefore proves equivalence even for partial trees, and only the selected search route needs insertion work.There is a distinction if “written as a fold” is required to mean a direct same-result-type expression Here
insert q = foldTree node base, with no richer result and final projection. On finite fully defined binary search trees, such an expression is possible by undoing insertion of a new extreme key in the child that should remain unchanged. For completeness, a literal direct fold on that finite, fully defined domain is:eraseFresh :: Ord a => a -> Tree a -> Tree a
eraseFresh _ Empty = Empty
eraseFresh q (Tree l x r) = case compare q x of
LT -> Tree (eraseFresh q l) x r
EQ -> Empty
GT -> Tree l x (eraseFresh q r)
insertDirectFold :: Ord a => a -> Tree a -> Tree a
insertDirectFold q = foldTree node (Tree Empty q Empty)
where
node left x right = case compare q x of
LT -> Tree left x (eraseFresh q right)
EQ -> Tree (eraseFresh q left) x (eraseFresh q right)
GT -> Tree (eraseFresh q left) x righteraseFresh is used only to undo a newly inserted extreme key, which is a leaf; it is not a general deletion operation. The binary search tree bound at each root guarantees exactly that precondition on every child being restored.On all partial Haskell trees, however, a direct expression with result type
Tree a cannot implement part (f).For that obstruction, fix Both can be completed to valid right subtrees of a root labelled
q = 0, and consider the two right subtreesr1 = Tree Empty undefined Empty
r2 = Tree Empty undefined (Tree Empty 3 Empty)1. Yet insert 0 r1 and insert 0 r2 are both bottom, because insertion must compare 0 with their undefined root label. A direct fold equal to insertion would therefore receive the same folded left value, root 1, and bottom folded right value on both larger trees Tree Empty 1 r1 and Tree Empty 1 r2. It must return the same result. Part (f) instead preserves r1 or r2, which are distinguishable by inspecting their right-child constructors without inspecting the undefined labels. This proves the direct-fold obstruction. The pair-valued implementation avoids losing that information and states precisely which sense of fold implementation is being used. Predicate 2026-10-25
Undoing insertion of a new extreme key 2026-10-24
If a key is smaller than every key of a finite binary search tree , insert and then delete the minimum: the original tree, including its shape, is recovered. The insertion follows the leftmost search route and creates a new minimum leaf; deleting that leaf reverses the same route. The symmetric statement uses a fresh maximum. At a node with root , the bounds on its children therefore let a direct tree fold reconstruct an untouched child from that child's insertion result. This gives a direct fold implementation of insertion without assuming that insertion is injective on arbitrary trees.