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Joined 2026-09-21 Message
Incoming links: Difference list
Linear flattening of a binary tree 2026-10-24
For a leaf-labelled binary tree, define
go (One x) acc = x:acc and go (Two l r) acc = go l (go r acc), then apply go t []. The invariant in the title follows by structural induction and associativity of list concatenation. It proves preservation of leaf order. Each leaf supplies one output constructor and each internal node requires constant bookkeeping, giving linear time in the number of nodes. By contrast, flattening both subtrees and concatenating their results can copy prefixes of sizes on a left-branching tree, taking quadratic time. The accumulator is the already-known suffix in a difference list representation. List concatenation 2026-10-24
Append the second list after the first. For a finite left list of length , consuming the concatenation costs to copy its spine and can share the second argument. Repeatedly appending large left lists can cause quadratic time complexity; difference lists make concatenation into function composition.
Pass an already constructed suffix through the traversal:The invariant is that
convert :: Expr -> RPolish
convert e = go e []
where
go :: Expr -> RPolish -> RPolish
go (Num n) rest = Number n : rest
go (App op left right) rest =
go left (go right (Operator op : rest))go e rest is the reverse Polish notation of e followed by rest. It holds at a numeric leaf. At an application, the result is the left subtree's tokens, then the right subtree's tokens, then the operator, then rest, exactly as required.Each vertex of the abstract syntax tree creates one output list cell and performs constant local work. Thus producing the complete result takes time for an expression with vertices. No repeated list concatenation copies a previously traversed subtree. This suffix-passing construction is a difference list traversal.
Carry the eventual output tail as an accumulator, thereby avoiding repeated list concatenation:
abs :: CatList a -> [a]
abs t = flatten t []
where
flatten Nil zs = zs
flatten (One x) zs = x : zs
flatten (Cat l r) zs = flatten l (flatten r zs)The invariant is
flatten t zs = denotation t ++ zs. It is immediate for Nil and One; at Cat, applying the two induction hypotheses and associativity of list concatenation proves it. In particular, the empty accumulator gives the required abstraction.Each concatenation tree constructor is processed once, and each
One emits one result cons. Consequently full production takes time, where is the input constructor count, on every tree shape. This is a difference list calculation written with an explicit tail, an instance of linear flattening of a binary tree. The call stack can have depth for a badly skewed input; a linear time bound does not imply logarithmic stack space.Let be the number of leaves; the total number of nodes is . List concatenation copies its left operand. For a tree whose left subtree contains leaves at each successive internal node, these copied lengths are . Their sum is , so the worst-case running time is .
A difference list representation avoids copying prefixes:
flattenFast :: Tree a -> [a]
flattenFast t = go t []
where
go (One x) acc = x:acc
go (Two l r) acc = go l (go r acc)The recursion invariant is
go t acc = flatten t ++ acc. The leaf case is immediate; the internal-node case uses associativity of list concatenation. Each node is visited once and each leaf contributes one output constructor, giving running time when the result is fully consumed. This is linear flattening of a binary tree and does not assume balance.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. Tree fold Created 2026-10-24 Updated 2026-10-31
A structural recursion replacing the empty binary tree by a value and a nonempty node by a combining function on its two folded children and root label. The result type may itself be a function type, allowing a fold to construct a membership predicate or a difference list rather than traversing all branches for every subsequent query.
More generally, choose one operation for each data constructor. For a leaf-labelled binary tree represented by
One a | Two (Tree a) (Tree a), the tree fold instead has type signature (a -> b) -> (b -> b -> b) -> Tree a -> b: one operation handles the leaf label, and the other combines the two folded children. Replacing the operations by the original data constructors reconstructs the input, as structural induction proves.