B-Trees Data Structure: A Comprehensive Guide with Implementation

A B-tree is a self-balancing tree data structure that maintains sorted data and allows for efficient insertion, deletion, and search operations. It is particularly well-suited for storage systems that read and write relatively large blocks of data, such as databases and file systems.

Properties of B-Trees:

Degree (Order): A B-tree is defined by a minimum degree t (where t ≥ 2)
Node Structure:
- Every node has at most 2t - 1 keys
- Every node (except root) has at least t - 1 keys
- The root may have as few as 1 key if it’s not a leaf
Height Balance: All leaves are at the same level
Key Distribution:
- Keys in a node are stored in sorted order
- Between two keys k₁ and k₂ in an internal node, all keys in the subtree must be greater than k₁ and less than k₂

Basic Operations on B-Trees

1. Search Operation

Similar to binary search tree but generalized for multiple keys per node
Time complexity: O(logₜ n) where t is the minimum degree

2. Insertion Operation

Always inserted at leaf level
May require splitting nodes to maintain B-tree properties
Time complexity: O(logₜ n)

3. Deletion Operation

More complex than insertion
May require borrowing from siblings or merging nodes
Time complexity: O(logₜ n)

Deleting a Key from a B-Tree

Deletion from a B-tree is more involved than insertion because we must ensure the tree remains balanced and all B-tree properties are maintained. Here’s the detailed process:

Cases for Deletion:

Key is in a leaf node:
- If the leaf has more than t-1 keys, simply remove the key
- If the leaf has exactly t-1 keys, we may need to:
  - Borrow a key from an immediate sibling (if sibling has more than t-1 keys)
  - Merge with a sibling (if both have t-1 keys)
Key is in an internal node:
- Replace the key with its predecessor (largest key in left subtree) or successor (smallest key in right subtree)
- Recursively delete the predecessor/successor (which will always be in a leaf)
Key is not present in the node where expected:
- If the child that would contain the key has only t-1 keys:
  - Borrow a key from an immediate sibling (if possible)
  - Merge with a sibling (if borrowing isn’t possible)
- Recursively proceed to the appropriate child

Deletion Algorithm Steps:

Start at the root
If the key is in the current node:
- If it’s a leaf node, remove the key
- If it’s an internal node:
  a. Find predecessor or successor
  b. Replace the key with the predecessor/successor
  c. Recursively delete the predecessor/successor
If the key is not in the current node:
- Find the child that would contain the key
- Ensure this child has at least t keys (borrow or merge if necessary)
- Recursively proceed to the appropriate child

C Implementation of B-Tree with Deletion

#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>

#define MIN_DEGREE 3
#define MAX_KEYS (2*MIN_DEGREE-1)
#define MIN_KEYS (MIN_DEGREE-1)

typedef struct BTreeNode {
    int keys[MAX_KEYS];
    struct BTreeNode *children[MAX_KEYS + 1];
    int num_keys;
    bool is_leaf;
} BTreeNode;

<em>// Create a new empty node</em>
BTreeNode* create_node(bool is_leaf) {
    BTreeNode* node = (BTreeNode*)malloc(sizeof(BTreeNode));
    node->num_keys = 0;
    node->is_leaf = is_leaf;
    for (int i = 0; i < MAX_KEYS + 1; i++) {
        node->children[i] = NULL;
    }
    return node;
}

<em>// Search for a key in the tree</em>
bool search(BTreeNode* root, int key) {
    int i = 0;
    while (i < root->num_keys && key > root->keys[i]) {
        i++;
    }
    
    if (i < root->num_keys && key == root->keys[i]) {
        return true;
    }
    
    if (root->is_leaf) {
        return false;
    }
    
    return search(root->children[i], key);
}

<em>// Insert a key into the tree (non-full node)</em>
void insert_non_full(BTreeNode* node, int key) {
    int i = node->num_keys - 1;
    
    if (node->is_leaf) {
        while (i >= 0 && key < node->keys[i]) {
            node->keys[i + 1] = node->keys[i];
            i--;
        }
        node->keys[i + 1] = key;
        node->num_keys++;
    } else {
        while (i >= 0 && key < node->keys[i]) {
            i--;
        }
        i++;
        
        if (node->children[i]->num_keys == MAX_KEYS) {
            split_child(node, i);
            if (key > node->keys[i]) {
                i++;
            }
        }
        insert_non_full(node->children[i], key);
    }
}

<em>// Split a full child of a node</em>
void split_child(BTreeNode* parent, int child_index) {
    BTreeNode* child = parent->children[child_index];
    BTreeNode* new_child = create_node(child->is_leaf);
    new_child->num_keys = MIN_KEYS;
    
    <em>// Copy the right half of keys to the new child</em>
    for (int i = 0; i < MIN_KEYS; i++) {
        new_child->keys[i] = child->keys[i + MIN_DEGREE];
    }
    
    <em>// Copy the right half of children if not leaf</em>
    if (!child->is_leaf) {
        for (int i = 0; i < MIN_DEGREE; i++) {
            new_child->children[i] = child->children[i + MIN_DEGREE];
        }
    }
    
    child->num_keys = MIN_KEYS;
    
    <em>// Make space for the new child in the parent</em>
    for (int i = parent->num_keys; i > child_index; i--) {
        parent->children[i + 1] = parent->children[i];
    }
    parent->children[child_index + 1] = new_child;
    
    <em>// Move the middle key of child to parent</em>
    for (int i = parent->num_keys - 1; i >= child_index; i--) {
        parent->keys[i + 1] = parent->keys[i];
    }
    parent->keys[child_index] = child->keys[MIN_KEYS];
    parent->num_keys++;
}

<em>// Insert a key into the tree</em>
void insert(BTreeNode** root, int key) {
    BTreeNode* node = *root;
    
    if (node->num_keys == MAX_KEYS) {
        BTreeNode* new_root = create_node(false);
        new_root->children[0] = node;
        *root = new_root;
        split_child(new_root, 0);
        insert_non_full(new_root, key);
    } else {
        insert_non_full(node, key);
    }
}

<em>// Find the predecessor of a key in a node</em>
int get_predecessor(BTreeNode* node, int index) {
    BTreeNode* current = node->children[index];
    while (!current->is_leaf) {
        current = current->children[current->num_keys];
    }
    return current->keys[current->num_keys - 1];
}

<em>// Find the successor of a key in a node</em>
int get_successor(BTreeNode* node, int index) {
    BTreeNode* current = node->children[index + 1];
    while (!current->is_leaf) {
        current = current->children[0];
    }
    return current->keys[0];
}

<em>// Merge a child with its sibling</em>
void merge(BTreeNode* node, int index) {
    BTreeNode* child = node->children[index];
    BTreeNode* sibling = node->children[index + 1];
    
    <em>// Bring the key from the parent down</em>
    child->keys[MIN_KEYS] = node->keys[index];
    
    <em>// Copy keys from sibling</em>
    for (int i = 0; i < sibling->num_keys; i++) {
        child->keys[i + MIN_DEGREE] = sibling->keys[i];
    }
    
    <em>// Copy children from sibling if not leaf</em>
    if (!child->is_leaf) {
        for (int i = 0; i <= sibling->num_keys; i++) {
            child->children[i + MIN_DEGREE] = sibling->children[i];
        }
    }
    
    <em>// Move keys in parent</em>
    for (int i = index + 1; i < node->num_keys; i++) {
        node->keys[i - 1] = node->keys[i];
    }
    
    <em>// Move children in parent</em>
    for (int i = index + 2; i <= node->num_keys; i++) {
        node->children[i - 1] = node->children[i];
    }
    
    child->num_keys += sibling->num_keys + 1;
    node->num_keys--;
    
    free(sibling);
}

<em>// Borrow a key from the left sibling</em>
void borrow_from_left(BTreeNode* node, int index) {
    BTreeNode* child = node->children[index];
    BTreeNode* sibling = node->children[index - 1];
    
    <em>// Move all keys in child right by 1</em>
    for (int i = child->num_keys - 1; i >= 0; i--) {
        child->keys[i + 1] = child->keys[i];
    }
    
    <em>// Move all children right by 1 if not leaf</em>
    if (!child->is_leaf) {
        for (int i = child->num_keys; i >= 0; i--) {
            child->children[i + 1] = child->children[i];
        }
    }
    
    <em>// Move key from parent to child</em>
    child->keys[0] = node->keys[index - 1];
    
    <em>// Move last key from sibling to parent</em>
    node->keys[index - 1] = sibling->keys[sibling->num_keys - 1];
    
    <em>// Move last child from sibling if not leaf</em>
    if (!child->is_leaf) {
        child->children[0] = sibling->children[sibling->num_keys];
    }
    
    child->num_keys++;
    sibling->num_keys--;
}

<em>// Borrow a key from the right sibling</em>
void borrow_from_right(BTreeNode* node, int index) {
    BTreeNode* child = node->children[index];
    BTreeNode* sibling = node->children[index + 1];
    
    <em>// Move key from parent to child</em>
    child->keys[child->num_keys] = node->keys[index];
    
    <em>// Move first key from sibling to parent</em>
    node->keys[index] = sibling->keys[0];
    
    <em>// Move first child from sibling if not leaf</em>
    if (!child->is_leaf) {
        child->children[child->num_keys + 1] = sibling->children[0];
    }
    
    <em>// Move all keys in sibling left by 1</em>
    for (int i = 1; i < sibling->num_keys; i++) {
        sibling->keys[i - 1] = sibling->keys[i];
    }
    
    <em>// Move all children left by 1 if not leaf</em>
    if (!sibling->is_leaf) {
        for (int i = 1; i <= sibling->num_keys; i++) {
            sibling->children[i - 1] = sibling->children[i];
        }
    }
    
    child->num_keys++;
    sibling->num_keys--;
}

<em>// Fill a child that has less than t-1 keys</em>
void fill(BTreeNode* node, int index) {
    <em>// If previous child has more than t-1 keys, borrow from it</em>
    if (index != 0 && node->children[index - 1]->num_keys >= MIN_DEGREE) {
        borrow_from_left(node, index);
    }
    <em>// If next child has more than t-1 keys, borrow from it</em>
    else if (index != node->num_keys && node->children[index + 1]->num_keys >= MIN_DEGREE) {
        borrow_from_right(node, index);
    }
    <em>// Otherwise, merge with a sibling</em>
    else {
        if (index != node->num_keys) {
            merge(node, index);
        } else {
            merge(node, index - 1);
        }
    }
}

<em>// Delete a key from a subtree</em>
void delete_from_subtree(BTreeNode* node, int key) {
    int index = 0;
    
    <em>// Find the key in the node or the child to descend to</em>
    while (index < node->num_keys && key > node->keys[index]) {
        index++;
    }
    
    <em>// Case 1: Key is present in this node</em>
    if (index < node->num_keys && key == node->keys[index]) {
        if (node->is_leaf) {
            <em>// Case 1a: Key is in a leaf node</em>
            for (int i = index + 1; i < node->num_keys; i++) {
                node->keys[i - 1] = node->keys[i];
            }
            node->num_keys--;
        } else {
            <em>// Case 1b: Key is in an internal node</em>
            if (node->children[index]->num_keys >= MIN_DEGREE) {
                <em>// Case 1b-i: Left child has at least t keys</em>
                int predecessor = get_predecessor(node, index);
                node->keys[index] = predecessor;
                delete_from_subtree(node->children[index], predecessor);
            } else if (node->children[index + 1]->num_keys >= MIN_DEGREE) {
                <em>// Case 1b-ii: Right child has at least t keys</em>
                int successor = get_successor(node, index);
                node->keys[index] = successor;
                delete_from_subtree(node->children[index + 1], successor);
            } else {
                <em>// Case 1b-iii: Both children have t-1 keys</em>
                merge(node, index);
                delete_from_subtree(node->children[index], key);
            }
        }
    } else {
        <em>// Case 2: Key is not present in this node</em>
        if (node->is_leaf) {
            printf("Key %d not found in the tree.\n", key);
            return;
        }
        
        bool is_last_child = (index == node->num_keys);
        
        if (node->children[index]->num_keys < MIN_DEGREE) {
            fill(node, index);
        }
        
        <em>// If the last child was merged, it must have merged with the previous child</em>
        if (is_last_child && index > node->num_keys) {
            delete_from_subtree(node->children[index - 1], key);
        } else {
            delete_from_subtree(node->children[index], key);
        }
    }
}

<em>// Delete a key from the tree</em>
void delete_key(BTreeNode** root, int key) {
    if (!*root) {
        printf("Tree is empty.\n");
        return;
    }
    
    delete_from_subtree(*root, key);
    
    <em>// If the root has no keys and has a child, make the child the new root</em>
    if ((*root)->num_keys == 0) {
        BTreeNode* old_root = *root;
        if ((*root)->is_leaf) {
            *root = NULL;
        } else {
            *root = (*root)->children[0];
        }
        free(old_root);
    }
}

<em>// Print the tree (inorder traversal)</em>
void print_tree(BTreeNode* node, int level) {
    if (node) {
        printf("Level %d: ", level);
        for (int i = 0; i < node->num_keys; i++) {
            printf("%d ", node->keys[i]);
        }
        printf("\n");
        
        if (!node->is_leaf) {
            for (int i = 0; i <= node->num_keys; i++) {
                print_tree(node->children[i], level + 1);
            }
        }
    }
}

<em>// Main function to test the B-tree implementation</em>
int main() {
    BTreeNode* root = create_node(true);
    
    int keys[] = {10, 20, 5, 6, 12, 30, 7, 17, 3, 1, 8, 25, 27, 28, 35, 40, 45};
    int n = sizeof(keys)/sizeof(keys[0]);
    
    for (int i = 0; i < n; i++) {
        insert(&root, keys[i]);
        printf("Inserted %d:\n", keys[i]);
        print_tree(root, 0);
        printf("\n");
    }
    
    printf("Final tree:\n");
    print_tree(root, 0);
    printf("\n");
    
    <em>// Delete some keys</em>
    int keys_to_delete[] = {6, 13, 7, 4, 20, 17, 10};
    int m = sizeof(keys_to_delete)/sizeof(keys_to_delete[0]);
    
    for (int i = 0; i < m; i++) {
        printf("Deleting %d:\n", keys_to_delete[i]);
        delete_key(&root, keys_to_delete[i]);
        print_tree(root, 0);
        printf("\n");
    }
    
    printf("Final tree after deletions:\n");
    print_tree(root, 0);
    
    return 0;
}

Explanation of the Implementation

Node Structure:
- Each node contains an array of keys and an array of child pointers
- num_keys tracks how many keys are currently in the node
- is_leaf indicates whether the node is a leaf
Insertion:
- The tree grows from the root when it becomes full
- insert_non_full handles inserting into nodes that aren’t full
- split_child splits a full child node during insertion
Deletion:
- delete_from_subtree handles the recursive deletion process
- fill ensures a node has enough keys before descending
- borrow_from_left and borrow_from_right handle key redistribution
- merge combines two nodes when redistribution isn’t possible
Helper Functions:
- get_predecessor and get_successor find replacement keys for internal node deletions
- print_tree displays the tree structure for visualization

This implementation maintains all B-tree properties during insertions and deletions, ensuring the tree remains balanced at all times. The minimum degree t is set to 3 in this example, but can be adjusted as needed.

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