BST Variants - Learning Module

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BST with Duplicate Handling

The Duplicate Key Dilemma

Throughout our exploration of Binary Search Trees, we've implicitly assumed that every key in the tree is unique. This assumption simplifies our understanding of the BST property: for every node, all keys in the left subtree are strictly less than the node's key, and all keys in the right subtree are strictly greater than the node's key.

But what happens when we need to store duplicate keys? This isn't an edge case—it's a fundamental requirement in countless real-world applications:

A priority queue where multiple tasks share the same priority
A database index on a non-unique column (e.g., customer ages, order dates)
An event log where multiple events share the same timestamp
A leaderboard where players can have identical scores

The moment we introduce duplicates, our clean BST property faces an existential question: Where does a duplicate go? Left? Right? Does it matter?

What You Will Master

By the end of this page, you will understand multiple strategies for handling duplicate keys in BSTs, evaluate their tradeoffs in terms of complexity, space, and use cases, and implement production-ready solutions. You'll learn why 'just pick left or right' isn't sufficient and how different strategies impact tree balance, search semantics, and deletion correctness.

The Fundamental Problem

Let's precisely define the problem. The standard BST property states:

For every node N with key k:

All keys in the left subtree of N are < k

All keys in the right subtree of N are > k

This definition uses strict inequalities. When we insert a duplicate key (a new node with key = k where k already exists), we have several questions to answer:

Where does it physically go? Left subtree, right subtree, or attached to the existing node?
Does our modified property remain searchable? Can we still find all occurrences efficiently?
How does deletion work? When removing a key that exists multiple times, what happens?
Does this affect tree balance? Can our choice cause pathological imbalance?

These questions reveal that duplicate handling isn't a minor detail—it's a design decision that permeates every BST operation.

The Naive Approach Fails

Simply saying 'put duplicates on the left' or 'put duplicates on the right' without careful analysis leads to subtle bugs. Consider: if duplicates always go right, repeated insertion of the same key creates a linked list hanging off that node—O(n) degeneracy from a single repeated value!

The semantic question:

Beyond implementation, we must consider what our BST means with duplicates:

Should search(k) return any node with key k, or a specific one?
Should we be able to count how many times k appears?
Is there an ordering among duplicates (e.g., are they maintained in insertion order)?
When we delete k, do we delete all occurrences or just one?

Different applications need different answers. A database index needs to find all matching records. A priority queue only needs to extract one minimum-priority item. A multiset needs accurate counts. Our strategy must match our use case.

Strategy 1: Allow Left or Right Placement

The most straightforward approach is to modify our BST property to accommodate duplicates. We have two variants:

Variant A: Duplicates go left

For every node N with key k:
- All keys in the left subtree are ≤ k
- All keys in the right subtree are > k

Variant B: Duplicates go right

For every node N with key k:
- All keys in the left subtree are < k
- All keys in the right subtree are ≥ k

Both variants are valid and maintain the searchability of the BST. The key insight is that we've replaced one strict inequality with a non-strict one, creating a 'direction' for duplicates to flow.

bst-duplicates-right.ts
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interface BSTNode<T> {
    key: number;
    value: T;
    left: BSTNode<T> | null;
    right: BSTNode<T> | null;
}
 
/**
 * Insert with duplicates going right.
 * BST Property: left < key, right >= key
 * 
 * This means when we encounter an equal key, we continue
 * searching in the right subtree until we find the correct
 * leaf position.
 */
function insertWithDuplicatesRight<T>(
    root: BSTNode<T> | null,
    key: number,
    value: T
): BSTNode<T> {
    if (root === null) {
        return { key, value, left: null, right: null };
    }
 
    if (key < root.key) {
        // Strictly less: go left
        root.left = insertWithDuplicatesRight(root.left, key, value);
    } else {
        // Greater OR EQUAL: go right
        // This is where duplicates get placed
        root.right = insertWithDuplicatesRight(root.right, key, value);
    }
 
    return root;
}
 
/**
 * Search for ANY node with the given key.
 * Returns the first match found (not necessarily the first inserted).
 */
function searchAny<T>(
    root: BSTNode<T> | null,
    key: number
): BSTNode<T> | null {
    if (root === null) return null;
 
    if (key === root.key) {
        return root; // Found one occurrence
    } else if (key < root.key) {
        return searchAny(root.left, key);
    } else {
        return searchAny(root.right, key);
    }
}
 
/**
 * Find ALL nodes with the given key.
 * With duplicates-right, after finding a match, more matches
 * can exist in the right subtree.
 */
function searchAll<T>(
    root: BSTNode<T> | null,
    key: number,
    results: BSTNode<T>[] = []
): BSTNode<T>[] {
    if (root === null) return results;
 
    if (key === root.key) {
        results.push(root);
        // More duplicates may exist in the right subtree
        searchAll(root.right, key, results);
    } else if (key < root.key) {
        searchAll(root.left, key, results);
    } else {
        searchAll(root.right, key, results);
    }
 
    return results;
}

Analysis of this approach:

Aspect	Evaluation
Insertion	O(h) - straightforward traversal
Search (any)	O(h) - stops at first match
Search (all)	O(h + k) - where k is number of duplicates
Deletion	Complex - must handle chains of duplicates
Space	One node per key occurrence
Balance	Risk of degenerate chains for repeated keys

Critical Weakness: Balance Degradation

If we insert the same key n times, all n nodes form a chain in one direction. For 'duplicates-right', inserting key=5 ten times creates a linked list of 10 nodes hanging to the right. This transforms our O(log n) tree into O(n) for those operations touching the duplicate chain. In extreme cases, this can devastate performance.

Strategy 2: Count Field per Node

A more space-efficient and balance-friendly approach is to augment each node with a count field that tracks how many times the key appears. Instead of creating new nodes for duplicates, we simply increment the counter.

Modified node structure:

Node {
    key: K
    value: V        // or a list of values if values differ
    count: number   // number of occurrences of this key
    left: Node | null
    right: Node | null
}

This approach maintains the original strict BST property (left < root < right) while supporting duplicates through the count mechanism.

bst-count-field.ts
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interface CountBSTNode<T> {
    key: number;
    values: T[];    // All values associated with this key
    count: number;  // Number of occurrences
    left: CountBSTNode<T> | null;
    right: CountBSTNode<T> | null;
}
 
/**
 * Insert with count tracking.
 * Maintains strict BST property: left < key < right
 * Duplicates increase the count rather than creating new nodes.
 */
function insertWithCount<T>(
    root: CountBSTNode<T> | null,
    key: number,
    value: T
): CountBSTNode<T> {
    if (root === null) {
        return {
            key,
            values: [value],
            count: 1,
            left: null,
            right: null,
        };
    }
 
    if (key === root.key) {
        // Duplicate found: increment count, store value
        root.count++;
        root.values.push(value);
    } else if (key < root.key) {
        root.left = insertWithCount(root.left, key, value);
    } else {
        root.right = insertWithCount(root.right, key, value);
    }
 
    return root;
}
 
/**
 * Search returns the node with count information.
 * Caller can see how many duplicates exist.
 */
function searchWithCount<T>(
    root: CountBSTNode<T> | null,
    key: number
): { found: boolean; count: number; values: T[] } {
    if (root === null) {
        return { found: false, count: 0, values: [] };
    }
 
    if (key === root.key) {
        return { found: true, count: root.count, values: root.values };
    } else if (key < root.key) {
        return searchWithCount(root.left, key);
    } else {
        return searchWithCount(root.right, key);
    }
}
 
/**
 * Delete one occurrence of a key.
 * Only removes node when count reaches zero.
 */
function deleteOne<T>(
    root: CountBSTNode<T> | null,
    key: number
): CountBSTNode<T> | null {
    if (root === null) return null;
 
    if (key < root.key) {
        root.left = deleteOne(root.left, key);
    } else if (key > root.key) {
        root.right = deleteOne(root.right, key);
    } else {
        // Found the key
        if (root.count > 1) {
            // Multiple occurrences: just decrement
            root.count--;
            root.values.pop(); // Remove last value (LIFO)
            return root;
        }
        
        // count === 1: perform actual node deletion
        // Standard BST deletion logic follows
        if (root.left === null) return root.right;
        if (root.right === null) return root.left;
 
        // Two children: find inorder successor
        let successor = root.right;
        while (successor.left !== null) {
            successor = successor.left;
        }
 
        root.key = successor.key;
        root.values = successor.values;
        root.count = successor.count;
        root.right = deleteOne(root.right, successor.key);
    }
 
    return root;
}

Count Field Approach: Comprehensive Analysis
Operation	Complexity	Notes
Insert (new key)	O(h)	Standard BST insertion
Insert (duplicate)	O(h)	Find node, increment count
Search	O(h)	Returns count and all values
Delete one	O(h)	Decrement count or remove node
Delete all of key	O(h)	Just remove the node
Count of key k	O(h)	Direct access via count field
Total elements	O(n)	Sum all counts in tree

Balance Preservation

The count approach is balance-friendly. Inserting the same key 1,000 times doesn't create 1,000 nodes—it creates one node with count=1000. This means duplicate-heavy workloads don't degrade tree structure. Combined with self-balancing (AVL, Red-Black), this approach guarantees O(log n) operations where n is the number of distinct keys.

When to use the count field approach:

✅ When you need to know how many times a key appears
✅ When values associated with duplicate keys are identical or can be aggregated
✅ When you want to preserve tree balance regardless of duplicate frequency
✅ For implementing multisets (bags) where multiplicity matters

When to avoid:

❌ When each duplicate has distinct associated data that must be preserved separately
❌ When insertion order among duplicates matters
❌ When you need to delete specific occurrences by criteria other than key

Strategy 3: List of Values per Node

A natural extension of the count field approach is to store all values associated with a key in a list. This is the most flexible solution and is commonly used in database indexes where a single index key (e.g., 'age=25') maps to multiple records.

Key insight: The BST is indexed by key, but each key maps to a collection of values rather than a single value. The tree structure remains strictly a BST, but the 'value' at each node is a list.

bst-value-list.ts
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interface ListBSTNode<T> {
    key: number;
    values: T[];  // All values with this key, preserving insertion order
    left: ListBSTNode<T> | null;
    right: ListBSTNode<T> | null;
}
 
/**
 * A BST-backed multimap: keys to lists of values.
 * Perfect for database indexes on non-unique columns.
 */
class BSTMultiMap<T> {
    private root: ListBSTNode<T> | null = null;
    private totalCount = 0;  // Total values across all keys
 
    /**
     * Insert a key-value pair.
     * If key exists, appends value to existing list.
     */
    insert(key: number, value: T): void {
        this.root = this.insertNode(this.root, key, value);
        this.totalCount++;
    }
 
    private insertNode(
        node: ListBSTNode<T> | null,
        key: number,
        value: T
    ): ListBSTNode<T> {
        if (node === null) {
            return { key, values: [value], left: null, right: null };
        }
 
        if (key === node.key) {
            node.values.push(value);  // Append to existing list
        } else if (key < node.key) {
            node.left = this.insertNode(node.left, key, value);
        } else {
            node.right = this.insertNode(node.right, key, value);
        }
 
        return node;
    }
 
    /**
     * Get all values associated with a key.
     * Returns empty array if key not found.
     */
    get(key: number): T[] {
        const node = this.findNode(this.root, key);
        return node ? [...node.values] : [];  // Return copy
    }
 
    /**
     * Get the count of values for a specific key.
     */
    count(key: number): number {
        const node = this.findNode(this.root, key);
        return node ? node.values.length : 0;
    }
 
    /**
     * Remove a specific value from a key's list.
     * Uses deep equality check for objects.
     */
    removeValue(key: number, value: T): boolean {
        const node = this.findNode(this.root, key);
        if (!node) return false;
 
        const index = node.values.findIndex((v) => 
            this.deepEquals(v, value)
        );
 
        if (index === -1) return false;
 
        node.values.splice(index, 1);
        this.totalCount--;
 
        // If no values remain, remove the node entirely
        if (node.values.length === 0) {
            this.root = this.deleteNode(this.root, key);
        }
 
        return true;
    }
 
    /**
     * Range query: get all values where key is in [minKey, maxKey].
     * This is where BST shines over hash tables.
     */
    range(minKey: number, maxKey: number): { key: number; values: T[] }[] {
        const results: { key: number; values: T[] }[] = [];
        this.rangeQuery(this.root, minKey, maxKey, results);
        return results;
    }
 
    private rangeQuery(
        node: ListBSTNode<T> | null,
        minKey: number,
        maxKey: number,
        results: { key: number; values: T[] }[]
    ): void {
        if (node === null) return;
 
        // If current key > minKey, left subtree may have valid keys
        if (node.key > minKey) {
            this.rangeQuery(node.left, minKey, maxKey, results);
        }
 
        // If current key is in range, include it
        if (node.key >= minKey && node.key <= maxKey) {
            results.push({ key: node.key, values: [...node.values] });
        }
 
        // If current key < maxKey, right subtree may have valid keys
        if (node.key < maxKey) {
            this.rangeQuery(node.right, minKey, maxKey, results);
        }
    }
 
    private findNode(
        node: ListBSTNode<T> | null,
        key: number
    ): ListBSTNode<T> | null {
        if (node === null) return null;
        if (key === node.key) return node;
        if (key < node.key) return this.findNode(node.left, key);
        return this.findNode(node.right, key);
    }
 
    private deleteNode(
        node: ListBSTNode<T> | null,
        key: number
    ): ListBSTNode<T> | null {
        if (node === null) return null;
 
        if (key < node.key) {
            node.left = this.deleteNode(node.left, key);
        } else if (key > node.key) {
            node.right = this.deleteNode(node.right, key);
        } else {
            // Standard BST delete
            if (!node.left) return node.right;
            if (!node.right) return node.left;
 
            let successor = node.right;
            while (successor.left) successor = successor.left;
 
            node.key = successor.key;
            node.values = successor.values;
            node.right = this.deleteNode(node.right, successor.key);
        }
 
        return node;
    }
 
    private deepEquals(a: T, b: T): boolean {
        return JSON.stringify(a) === JSON.stringify(b);
    }
}

Database Index Pattern

This is essentially how database secondary indexes work on non-unique columns. An index on 'customer_age' creates a BST (often a B-tree variant) where each age value maps to a list of record IDs. A query like 'SELECT * WHERE age BETWEEN 25 AND 35' becomes an efficient range query on this structure.

Comparison with Strategy 2 (Count Field):

Aspect	Count Field	Value List
Memory per duplicate	Just +1 integer	+1 pointer/value
Access individual values	❌ Not stored	✅ Preserved
Insertion order	❌ Lost	✅ Maintained
Remove specific value	❌ Can only decrement	✅ Find and remove
Best for	Multiset (counting)	Multi-map (indexing)

Strategy 4: Linked List of Nodes

A hybrid approach chains duplicate nodes together using an additional pointer. Each node with a duplicate key links to the next node with the same key. This preserves individual node identity while avoiding the balance problems of Strategy 1.

Node structure:

Node {
    key: K
    value: V
    left: Node | null
    right: Node | null
    next: Node | null   // Link to next duplicate
}

The BST structure maintains strict ordering (left < root < right), and duplicates form a singly-linked list attached to the first occurrence.

bst-linked-duplicates.ts
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interface LinkedBSTNode<T> {
    key: number;
    value: T;
    left: LinkedBSTNode<T> | null;
    right: LinkedBSTNode<T> | null;
    next: LinkedBSTNode<T> | null;  // Chain of duplicates
}
 
/**
 * Insert with linked duplicate chain.
 * First occurrence is in tree structure; duplicates chain off it.
 */
function insertLinked<T>(
    root: LinkedBSTNode<T> | null,
    key: number,
    value: T
): LinkedBSTNode<T> {
    if (root === null) {
        return { key, value, left: null, right: null, next: null };
    }
 
    if (key === root.key) {
        // Found existing node with same key: chain the duplicate
        const newNode: LinkedBSTNode<T> = {
            key,
            value,
            left: null,
            right: null,
            next: root.next,  // Point to current chain
        };
        root.next = newNode;  // New node becomes head of chain
    } else if (key < root.key) {
        root.left = insertLinked(root.left, key, value);
    } else {
        root.right = insertLinked(root.right, key, value);
    }
 
    return root;
}
 
/**
 * Find all values for a key by traversing the chain.
 */
function findAllLinked<T>(
    root: LinkedBSTNode<T> | null,
    key: number
): T[] {
    const node = findNodeLinked(root, key);
    if (!node) return [];
 
    const results: T[] = [];
    let current: LinkedBSTNode<T> | null = node;
 
    // Traverse the chain
    while (current !== null) {
        results.push(current.value);
        current = current.next;
    }
 
    return results;
}
 
function findNodeLinked<T>(
    root: LinkedBSTNode<T> | null,
    key: number
): LinkedBSTNode<T> | null {
    if (root === null) return null;
    if (key === root.key) return root;
    if (key < root.key) return findNodeLinked(root.left, key);
    return findNodeLinked(root.right, key);
}
 
/**
 * Delete one occurrence of a key.
 * If it's the primary node and has a chain, promote next in chain.
 */
function deleteOneLinked<T>(
    root: LinkedBSTNode<T> | null,
    key: number
): LinkedBSTNode<T> | null {
    if (root === null) return null;
 
    if (key < root.key) {
        root.left = deleteOneLinked(root.left, key);
        return root;
    }
 
    if (key > root.key) {
        root.right = deleteOneLinked(root.right, key);
        return root;
    }
 
    // Found the key
    if (root.next !== null) {
        // Has duplicates in chain: promote next to current position
        const promoted = root.next;
        promoted.left = root.left;
        promoted.right = root.right;
        return promoted;
    }
 
    // No duplicates: standard BST delete
    if (root.left === null) return root.right;
    if (root.right === null) return root.left;
 
    // Two children: find inorder successor
    let successor = root.right;
    let successorParent = root;
 
    while (successor.left !== null) {
        successorParent = successor;
        successor = successor.left;
    }
 
    // Replace current with successor
    root.key = successor.key;
    root.value = successor.value;
    root.next = successor.next;
 
    if (successorParent === root) {
        root.right = successor.right;
    } else {
        successorParent.left = successor.right;
    }
 
    return root;
}

Advantages of the linked chain approach:

Tree structure preserved — Only unique keys occupy tree positions, so inserting 1000 duplicates doesn't add 1000 tree nodes
Individual node identity — Each insertion creates a distinct node with its own value
Efficient deletion — Removing a specific occurrence is O(h + k) where k is position in chain
Compatible with balancing — Since tree structure has one node per unique key, self-balancing works normally

Disadvantages:

Memory overhead — Extra pointer per node (8 bytes on 64-bit)
Chain traversal — Finding all duplicates requires O(k) additional operations
Complexity — More edge cases in deletion logic

When to Use Linked Chains

This approach shines when duplicates are common, each duplicate must be individually addressable, and you need to maintain compatibility with tree-balancing algorithms. It's used in some database implementations where each 'duplicate' entry has its own transaction metadata or version information.

Comparative Analysis: Choosing the Right Strategy

Let's synthesize our strategies into a decision framework. The right choice depends on your specific requirements:

Duplicate Handling Strategies: Complete Comparison
Strategy	Balance Impact	Space per Dup	Use Case
Left/Right Placement	❌ Can degrade to O(n)	Full node	Simple, few duplicates
Count Field	✅ No impact	1 integer	Counting, multiset
Value List	✅ No impact	1 pointer + value	Multi-map, indexing
Linked Chain	✅ Minimal impact	Full node + 1 ptr	Individual node identity

Decision flowchart:

Start
  │
  ├─ Do duplicates need individual identity?
  │    │
  │    ├─ Yes → Do you need to preserve tree balance?
  │    │         │
  │    │         ├─ Yes → Use Linked Chain (Strategy 4)
  │    │         │
  │    │         └─ No, duplicates are rare → Use Left/Right (Strategy 1)
  │    │
  │    └─ No → Is the value different for each duplicate?
  │              │
  │              ├─ Yes → Use Value List (Strategy 3)
  │              │
  │              └─ No → Use Count Field (Strategy 2)

Production Recommendation

For most production systems, start with Strategy 2 (Count Field) or Strategy 3 (Value List). These preserve tree balance, have excellent performance characteristics, and handle the majority of use cases. Only use Strategy 1 for quick prototypes or when duplicates are extremely rare. Use Strategy 4 when individual node identity is truly required.

Real-World Applications

Understanding duplicate handling isn't just academic—it directly impacts how you design systems. Let's examine concrete applications:

Database Secondary Indexes

•Problem: Index customers by age, where many customers share the same age
•Solution: Value List strategy (Strategy 3) — each age maps to list of customer IDs
•Why: Need to retrieve all matching records; range queries on ages are common
•Implementation: B-tree (self-balancing BST variant) with clustered lists at leaves

Event Log Indexing

•Problem: Index events by timestamp, where multiple events can share exact timestamps
•Solution: Linked Chain strategy (Strategy 4) — each event is individual node
•Why: Each event has unique metadata (type, payload) that must be preserved
•Benefit: Range queries ('events between 10am and 11am') remain O(log n + k)

Frequency Analysis (Bag/Multiset)

•Problem: Count word frequencies in a document where words repeat
•Solution: Count Field strategy (Strategy 2) — each word maps to its count
•Why: Only care about count, not individual occurrences
•Bonus: In-order traversal gives words sorted with their frequencies

Gaming Leaderboards

•Problem: Rank players by score where many players tie
•Solution: Value List strategy (Strategy 3) — each score maps to list of player IDs
•Why: Need to find all players at a given score, support range queries
•Extension: Augmented BST (covered next) to efficiently query rank

Summary: BST Duplicate Handling

We've thoroughly explored the duplicate handling problem in Binary Search Trees. Let's consolidate the key insights:

Key Takeaways

•Duplicates break the naive BST property — The strict left < root < right invariant must be modified or circumvented
•Four primary strategies exist — Left/Right placement, Count Field, Value List, and Linked Chain, each with distinct tradeoffs
•Balance is the critical concern — Naive approaches can degrade to O(n) with duplicate-heavy workloads
•Choose based on requirements — Do duplicates need identity? Is value data per duplicate? Is counting sufficient?
•Production systems prefer count/list approaches — They preserve balance and handle scale gracefully
•Real databases use these patterns — Secondary indexes, event stores, and analytics all leverage duplicate-aware BST variants

What's next:

Now that we understand duplicate handling, we're ready to explore more sophisticated BST extensions. The next page covers Order Statistics Trees—a powerful augmentation that adds rank-based operations to BSTs, enabling queries like 'find the k-th smallest element' in O(log n) time.

Page Complete

You now understand the full spectrum of duplicate handling strategies in BSTs. You can evaluate tradeoffs, implement production-ready solutions, and recognize which strategy fits your application requirements. This foundation prepares you for the augmented BST patterns that follow.