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Joined 2026-09-21 Message
Incoming links: Binary search tree
Perform a destructive binary search tree flattening by converting both child subtrees and splicing their endpoints around the original root:Save the child references before changing their interpretation. The input is a finite binary search tree satisfying part (b), and no external client may simultaneously rely on the old tree links. By structural induction, the two recursive calls return disjoint, ordered doubly linked lists containing exactly their original entries. The binary search tree bounds place every left word before
def flatten(t: Tree): (Tree, Tree) = {
if (t == null) (null, null)
else {
val oldLeft = t.left
val oldRight = t.right
val (ls, le) = flatten(oldLeft)
val (rs, re) = flatten(oldRight)
t.left = le
if (le != null) le.right = t
t.right = rs
if (rs != null) rs.left = t
val start = if (ls == null) t else ls
val end = if (re == null) t else re
(start, end)
}
}t.word and every right word after it. Splicing the left end, t, and the right start therefore preserves strict order and reciprocal links. A missing left list makes t.left null and t the start; a missing right list similarly makes t the end. The resulting outside links are null as required by part (i).The recursive loop variant analogue is the strictly smaller number of entries in each child subtree. Every entry is processed once and incurs a constant number of link writes, so time complexity is and recursion uses auxiliary space. No new
Tree entry is created; returning endpoint tuples and using a call stack are permitted auxiliary storage. The old binary search tree representation is consumed by this operation.start == null if and only if end == null. Otherwise there is a finite list of distinct software objects , with start eq v0, end eq v(m-1), v0.left == null, and v(m-1).right == null. For every valid adjacent pair, v(i).right eq v(i+1) and v(i+1).left eq v(i). These are exactly the entries reachable through the list links, so there is no extra cycle or branch. All counts are positive, all words are non-null, and words strictly increase from start to end.This is the representation invariant for a null-terminated doubly linked list without a list header. The abstraction function reads the ordered word-count pairs from left to right. Conversion additionally requires the output entries, counts and words to be precisely those of the input binary search tree; that preservation property is an operation postcondition, rather than an invariant of an arbitrary list by itself.
A minimum-oriented binary heap is an appropriate alternative. Store a complete binary tree in an array and maintain heap order. Insertion appends a value and moves it upward; removing the root replaces it by the last value and moves that value downward. Each path has levels, and equal values can be stored in separate slots. Both operations therefore take time when capacity is available.
Unlike a binary search tree, the heap does not maintain a complete search order, so finding an arbitrary specified value can require inspections. A simple array implementation also has a capacity issue: a fixed array can fill, and resizing a growable array occasionally costs , giving an amortized insertion guarantee unless allocation is otherwise managed. This capacity cost is an implementation tradeoff, not an inherent inability of heaps to represent the priority queue abstraction.
Not automatically: the duplicate-value policy must implement a multiset. A strict binary search tree storing one entry per distinct key represents a set; inserting a second equal value then loses a required occurrence. A fixed rule that puts all equal values on one side also needs care, because ordinary balancing rotations need not preserve that one-sided equality rule.
One valid refinement stores a multiplicity at each distinct key. Inserting an equal key increments its count;
delMin decrements the minimum key's count and removes that node only at zero. Alternatively, use distinct internal keys such as (value, uniqueSerial) while ordering primarily by value and returning only the value. With an appropriate balanced representation either approach preserves every occurrence and the required operation bounds. A bare claim about balanced search trees, with no treatment of duplicates, does not establish the abstract specification.Yes, provided balancing guarantees logarithmic height. In a suitably maintained balanced binary tree of height , insertion searches one root-to-leaf path and deletion of the minimum follows the leftmost path. Reconnecting the tree and performing the required rebalancing also costs at most a constant amount per level for the standard logarithmic-time schemes. Thus both operations take time, conventionally written for nontrivial sizes. Merely calling a tree approximately balanced would not prove that height bound; it is the balancing guarantee that matters. Without it, a binary search tree can have linear height.
The relevant quantity is the hash-table load factor , where is the number of distinct keys and the bucket count. With a suitable distribution of hashes, separate chaining has expected operation cost , treating key operations as constant-cost. Keeping only one hundred buckets while grows therefore increases average search length.
Balanced binary search trees could reduce a bucket search from linear time in its occupancy to logarithmic time. Unbalanced binary search trees provide no such guarantee: insertion order can produce a chain. Even balanced trees add comparison and node-management overhead, which may dominate for small buckets.
A usual first remedy is grow the table and rehash its entries, keeping the load factor bounded. Geometric growth gives amortized constant resizing overhead per insertion; with suitable hashing, expected lookup, update and insertion costs remain constant. Resizing is when performed and is not a worst-case constant-time guarantee for that particular operation. Balanced buckets can still be useful for heavy or adversarial collisions, but replacing every bucket tree does not by itself repair a poor hash function. For variable-length strings, hashing and string comparisons add their own character-processing costs.
Use the empty reference as the base case:Induction on the tree proves the order: the first recursive call prints exactly the left subtree, then the procedure prints the root, then the final call prints exactly the right subtree. A finite acyclic tree supplies decreasing proper subtrees, so the recursion terminates. The traversal is valid for arbitrary integer labels; sorted output would additionally require a binary search tree invariant.
def printInorder(root: Tree): Unit = {
if (root != null) {
printInorder(root.left)
println(root.datum)
printInorder(root.right)
}
}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.Choose the middle element as the root:The input is a finite, strictly ascending list, as required for distinct elements of a set. The two child lengths differ by at most one. Applying the same argument recursively gives a balanced binary tree of logarithmic height. All values on each side retain their original order, proving the binary search tree property. With linked lists, the repeated splitting gives time; it does not repeatedly insert values into a skewed tree.
build :: [a] -> Tree a
build [] = Empty
build xs = Tree (build ls) x (build rs)
where
(ls,x:rs) = splitAt (length xs `div` 2) xsFold into a membership predicate:At an empty tree the predicate is always false. At a nonempty binary search tree, an unequal query can occur only in the indicated child, while equality finds the root. This proves correctness inductively. Under lazy evaluation, the unused folded child is not demanded; lookup follows one root-to-leaf route and takes comparisons in a tree of height . Defining the predicate by a fold does not require visiting both subtrees for each query.
find :: Ord a => Tree a -> a -> Bool
find = foldTree node (const False)
where
node left x right q = case compare q x of
LT -> left q
EQ -> True
GT -> right qInsert down the unique search route, leaving equal elements unchanged:The binary search tree ordering determines the only child in which a missing query can belong. The recursive call preserves that child's ordering bounds, and the other child is shared. Thus the output represents the original set with
insert :: Ord a => a -> Tree a -> Tree a
insert q Empty = Tree Empty q Empty
insert q t@(Tree l x r) = case compare q x of
LT -> Tree (insert q l) x r
EQ -> t
GT -> Tree l x (insert q r)q added, with duplicates suppressed. Its time complexity and newly allocated route length are ; it makes no balancing guarantee.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. Persistent binary search tree deletion 2026-10-31
A pure function deletes a key from a binary search tree by rebuilding its search path and sharing untouched subtrees. At a matching node with two children, replace its key by the least key in its right subtree, removing that occurrence from the right subtree. The ordering invariant is preserved, with time complexity and fresh-node allocation proportional to the height of a binary tree.
Predicate 2026-10-25
Red-black tree 2026-10-31
A red-black tree colours each stored node red or black, treats absent-child sentinels as black, permits no red node to have a red child, and gives every path from a node to an absent-child sentinel the same number of black nodes. We also require a black root. These invariants bound its height of a binary tree by for stored keys: a path has at most twice as many real nodes as black nodes, and a subtree of black height has at least real nodes by mathematical induction. The structure therefore supports logarithmic-time directed binary search tree lookup.
Red-black tree insertion 2026-10-31
In the usual bottom-up algorithm, insert a red node by the binary search tree rule. If its parent is black, no repair is needed. If its parent and uncle are red, recolour both black and recolour the grandparent red, then continue upward. If the uncle is black, an inner configuration first rotates the parent to become an outer configuration; rotating the grandparent and exchanging its colour with the parent's resolves the red-red conflict. Apply the left/right mirror cases as needed and finish with a black root. The tree rotations preserve key order, and the colour changes preserve equal black height. Different valid insertion conventions can produce different valid red-black trees, so a worked trace must identify its convention.
Splaying 2026-10-31
In ordinary bottom-up splaying, locate the selected node by a binary search tree search and repeatedly move it toward the root. If its parent is the root, perform a zig step. Otherwise use a zig-zig step when the node and its parent occupy the same left/right directions, or a zig-zag step when their directions differ. Each tree rotation preserves the inorder traversal and hence the represented key order. The process ends with the selected key at the root. Unsuccessful access policies can instead move the last visited node to the root; arguments about successful calls should say the requested key exists.
Splay tree 2026-10-31
A splay tree is a binary search tree whose access operations use splaying to move a selected key to the root through tree rotations. It stores no explicit balance or height information. A single access can take linear time, while the standard operations have logarithmic amortized time complexity. The distinction matters: amortized analysis bounds the total cost of an operation sequence, not the worst cost of every individual operation.[ref]
Self-Adjusting Binary Search Trees by Daniel Sleator and Robert Tarjan. Author-hosted abstract: representation, operations and amortized logarithmic bound
Subtree Created 2026-10-13 Updated 2026-10-14
For a vertex in a rooted tree, its subtree contains and every vertex reachable by repeatedly following children in a rooted tree, together with the connecting edges; its root of a tree is . In a binary tree, the left subtree is rooted at the left child, and is empty when that child is absent; the right subtree is defined similarly. A binary search tree constrains every key in these subtrees.