B+ Tree
In this tutorial, you will learn what a B+ tree is. Also, you will find working examples of searching operation on a B+ tree in C.
A B+ tree is an advanced form of a self-balancing tree in which all the values are present in the leaf level.
An important concept to be understood before learning B+ tree is multilevel indexing. In multilevel indexing, the index of indices is created as in figure below. It makes accessing the data easier and faster.
Properties of a B+ Tree
- All leaves are at the same level.
- The root has at least two children.
- Each node except root can have a maximum of m children and at least m/2 children.
- Each node can contain a maximum of m – 1 keys and a minimum of ⌈m/2⌉ – 1 keys.
Comparison between a B-tree and a B+ Tree
The data pointers are present only at the leaf nodes on a B+ tree whereas the data pointers are present in the internal, leaf or root nodes on a B-tree.
The leaves are not connected with each other on a B-tree whereas they are connected on a B+ tree.
Operations on a B+ tree are faster than on a B-tree.
Searching on a B+ Tree
The following steps are followed to search for data in a B+ Tree of order m. Let the data to be searched be k.
- Start from the root node. Compare k with the keys at the root node [k1, k2, k3,……km – 1].
- If k < k1, go to the left child of the root node.
- Else if k == k1, compare k2. If
k < k2, k lies between k1 and k2. So, search in the left child of k2. - If k > k2, go for k3, k4,…km-1 as in steps 2 and 3.
- Repeat the above steps until a leaf node is reached.
- If k exists in the leaf node, return true else return false.
Searching Example on a B+ Tree
Let us search k = 45 on the following B+ tree.
- Compare k with the root node.
k is not found at the root - Since k > 25, go to the right child.
Go to right of the root - Compare k with 35. Since k > 30, compare k with 45.
k not found - Since k ≥ 45, so go to the right child.
go to the right - k is found.
k is found
C Examples
// Searching on a B+ Tree in C
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
// Default order
#define ORDER 3
typedef struct record {
int value;
} record;
// Node
typedef struct node {
void **pointers;
int *keys;
struct node *parent;
bool is_leaf;
int num_keys;
struct node *next;
} node;
int order = ORDER;
node *queue = NULL;
bool verbose_output = false;
// Enqueue
void enqueue(node *new_node);
// Dequeue
node *dequeue(void);
int height(node *const root);
int pathToLeaves(node *const root, node *child);
void printLeaves(node *const root);
void printTree(node *const root);
void findAndPrint(node *const root, int key, bool verbose);
void findAndPrintRange(node *const root, int range1, int range2, bool verbose);
int findRange(node *const root, int key_start, int key_end, bool verbose,
int returned_keys[], void *returned_pointers[]);
node *findLeaf(node *const root, int key, bool verbose);
record *find(node *root, int key, bool verbose, node **leaf_out);
int cut(int length);
record *makeRecord(int value);
node *makeNode(void);
node *makeLeaf(void);
int getLeftIndex(node *parent, node *left);
node *insertIntoLeaf(node *leaf, int key, record *pointer);
node *insertIntoLeafAfterSplitting(node *root, node *leaf, int key,
record *pointer);
node *insertIntoNode(node *root, node *parent,
int left_index, int key, node *right);
node *insertIntoNodeAfterSplitting(node *root, node *parent,
int left_index,
int key, node *right);
node *insertIntoParent(node *root, node *left, int key, node *right);
node *insertIntoNewRoot(node *left, int key, node *right);
node *startNewTree(int key, record *pointer);
node *insert(node *root, int key, int value);
// Enqueue
void enqueue(node *new_node){
node *c;
if (queue == NULL) {
queue = new_node;
queue->next = NULL;
} else {
c = queue;
while (c->next != NULL) {
c = c->next;
}
c->next = new_node;
new_node->next = NULL;
}
}
// Dequeue
node *dequeue(void){
node *n = queue;
queue = queue->next;
n->next = NULL;
return n;
}
// Print the leaves
void printLeaves(node *const root){
if (root == NULL) {
printf("Empty tree.n");
return;
}
int i;
node *c = root;
while (!c->is_leaf)
c = c->pointers[0];
while (true) {
for (i = 0; i < c->num_keys; i++) {
if (verbose_output)
printf("%p ", c->pointers[i]);
printf("%d ", c->keys[i]);
}
if (verbose_output)
printf("%p ", c->pointers[order - 1]);
if (c->pointers[order - 1] != NULL) {
printf(" | ");
c = c->pointers[order - 1];
} else
break;
}
printf("n");
}
// Calculate height
int height(node *const root){
int h = 0;
node *c = root;
while (!c->is_leaf) {
c = c->pointers[0];
h++;
}
return h;
}
// Get path to root
int pathToLeaves(node *const root, node *child){
int length = 0;
node *c = child;
while (c != root) {
c = c->parent;
length++;
}
return length;
}
// Print the tree
void printTree(node *const root){
node *n = NULL;
int i = 0;
int rank = 0;
int new_rank = 0;
if (root == NULL) {
printf("Empty tree.n");
return;
}
queue = NULL;
enqueue(root);
while (queue != NULL) {
n = dequeue();
if (n->parent != NULL && n == n->parent->pointers[0]) {
new_rank = pathToLeaves(root, n);
if (new_rank != rank) {
rank = new_rank;
printf("n");
}
}
if (verbose_output)
printf("(%p)", n);
for (i = 0; i < n->num_keys; i++) {
if (verbose_output)
printf("%p ", n->pointers[i]);
printf("%d ", n->keys[i]);
}
if (!n->is_leaf)
for (i = 0; i <= n->num_keys; i++)
enqueue(n->pointers[i]);
if (verbose_output) {
if (n->is_leaf)
printf("%p ", n->pointers[order - 1]);
else
printf("%p ", n->pointers[n->num_keys]);
}
printf("| ");
}
printf("n");
}
// Find the node and print it
void findAndPrint(node *const root, int key, bool verbose){
node *leaf = NULL;
record *r = find(root, key, verbose, NULL);
if (r == NULL)
printf("Record not found under key %d.n", key);
else
printf("Record at %p -- key %d, value %d.n",
r, key, r->value);
}
// Find and print the range
void findAndPrintRange(node *const root, int key_start, int key_end,
bool verbose){
int i;
int array_size = key_end - key_start + 1;
int returned_keys[array_size];
void *returned_pointers[array_size];
int num_found = findRange(root, key_start, key_end, verbose,
returned_keys, returned_pointers);
if (!num_found)
printf("None found.n");
else {
for (i = 0; i < num_found; i++)
printf("Key: %d Location: %p Value: %dn",
returned_keys[i],
returned_pointers[i],
((record *)
returned_pointers[i])
->value);
}
}
// Find the range
int findRange(node *const root, int key_start, int key_end, bool verbose,
int returned_keys[], void *returned_pointers[]){
int i, num_found;
num_found = 0;
node *n = findLeaf(root, key_start, verbose);
if (n == NULL)
return 0;
for (i = 0; i < n->num_keys && n->keys[i] < key_start; i++)
;
if (i == n->num_keys)
return 0;
while (n != NULL) {
for (; i < n->num_keys && n->keys[i] <= key_end; i++) {
returned_keys[num_found] = n->keys[i];
returned_pointers[num_found] = n->pointers[i];
num_found++;
}
n = n->pointers[order - 1];
i = 0;
}
return num_found;
}
// Find the leaf
node *findLeaf(node *const root, int key, bool verbose){
if (root == NULL) {
if (verbose)
printf("Empty tree.n");
return root;
}
int i = 0;
node *c = root;
while (!c->is_leaf) {
if (verbose) {
printf("[");
for (i = 0; i < c->num_keys - 1; i++)
printf("%d ", c->keys[i]);
printf("%d] ", c->keys[i]);
}
i = 0;
while (i < c->num_keys) {
if (key >= c->keys[i])
i++;
else
break;
}
if (verbose)
printf("%d ->n", i);
c = (node *)c->pointers[i];
}
if (verbose) {
printf("Leaf [");
for (i = 0; i < c->num_keys - 1; i++)
printf("%d ", c->keys[i]);
printf("%d] ->n", c->keys[i]);
}
return c;
}
record *find(node *root, int key, bool verbose, node **leaf_out){
if (root == NULL) {
if (leaf_out != NULL) {
*leaf_out = NULL;
}
return NULL;
}
int i = 0;
node *leaf = NULL;
leaf = findLeaf(root, key, verbose);
for (i = 0; i < leaf->num_keys; i++)
if (leaf->keys[i] == key)
break;
if (leaf_out != NULL) {
*leaf_out = leaf;
}
if (i == leaf->num_keys)
return NULL;
else
return (record *)leaf->pointers[i];
}
int cut(int length){
if (length % 2 == 0)
return length / 2;
else
return length / 2 + 1;
}
record *makeRecord(int value){
record *new_record = (record *)malloc(sizeof(record));
if (new_record == NULL) {
perror("Record creation.");
exit(EXIT_FAILURE);
} else {
new_record->value = value;
}
return new_record;
}
node *makeNode(void){
node *new_node;
new_node = malloc(sizeof(node));
if (new_node == NULL) {
perror("Node creation.");
exit(EXIT_FAILURE);
}
new_node->keys = malloc((order - 1) * sizeof(int));
if (new_node->keys == NULL) {
perror("New node keys array.");
exit(EXIT_FAILURE);
}
new_node->pointers = malloc(order * sizeof(void *));
if (new_node->pointers == NULL) {
perror("New node pointers array.");
exit(EXIT_FAILURE);
}
new_node->is_leaf = false;
new_node->num_keys = 0;
new_node->parent = NULL;
new_node->next = NULL;
return new_node;
}
node *makeLeaf(void){
node *leaf = makeNode();
leaf->is_leaf = true;
return leaf;
}
int getLeftIndex(node *parent, node *left){
int left_index = 0;
while (left_index <= parent->num_keys &&
parent->pointers[left_index] != left)
left_index++;
return left_index;
}
node *insertIntoLeaf(node *leaf, int key, record *pointer){
int i, insertion_point;
insertion_point = 0;
while (insertion_point < leaf->num_keys && leaf->keys[insertion_point] < key)
insertion_point++;
for (i = leaf->num_keys; i > insertion_point; i--) {
leaf->keys[i] = leaf->keys[i - 1];
leaf->pointers[i] = leaf->pointers[i - 1];
}
leaf->keys[insertion_point] = key;
leaf->pointers[insertion_point] = pointer;
leaf->num_keys++;
return leaf;
}
node *insertIntoLeafAfterSplitting(node *root, node *leaf, int key, record *pointer){
node *new_leaf;
int *temp_keys;
void **temp_pointers;
int insertion_index, split, new_key, i, j;
new_leaf = makeLeaf();
temp_keys = malloc(order * sizeof(int));
if (temp_keys == NULL) {
perror("Temporary keys array.");
exit(EXIT_FAILURE);
}
temp_pointers = malloc(order * sizeof(void *));
if (temp_pointers == NULL) {
perror("Temporary pointers array.");
exit(EXIT_FAILURE);
}
insertion_index = 0;
while (insertion_index < order - 1 && leaf->keys[insertion_index] < key)
insertion_index++;
for (i = 0, j = 0; i < leaf->num_keys; i++, j++) {
if (j == insertion_index)
j++;
temp_keys[j] = leaf->keys[i];
temp_pointers[j] = leaf->pointers[i];
}
temp_keys[insertion_index] = key;
temp_pointers[insertion_index] = pointer;
leaf->num_keys = 0;
split = cut(order - 1);
for (i = 0; i < split; i++) {
leaf->pointers[i] = temp_pointers[i];
leaf->keys[i] = temp_keys[i];
leaf->num_keys++;
}
for (i = split, j = 0; i < order; i++, j++) {
new_leaf->pointers[j] = temp_pointers[i];
new_leaf->keys[j] = temp_keys[i];
new_leaf->num_keys++;
}
free(temp_pointers);
free(temp_keys);
new_leaf->pointers[order - 1] = leaf->pointers[order - 1];
leaf->pointers[order - 1] = new_leaf;
for (i = leaf->num_keys; i < order - 1; i++)
leaf->pointers[i] = NULL;
for (i = new_leaf->num_keys; i < order - 1; i++)
new_leaf->pointers[i] = NULL;
new_leaf->parent = leaf->parent;
new_key = new_leaf->keys[0];
return insertIntoParent(root, leaf, new_key, new_leaf);
}
node *insertIntoNode(node *root, node *n,
int left_index, int key, node *right){
int i;
for (i = n->num_keys; i > left_index; i--) {
n->pointers[i + 1] = n->pointers[i];
n->keys[i] = n->keys[i - 1];
}
n->pointers[left_index + 1] = right;
n->keys[left_index] = key;
n->num_keys++;
return root;
}
node *insertIntoNodeAfterSplitting(node *root, node *old_node, int left_index,
int key, node *right){
int i, j, split, k_prime;
node *new_node, *child;
int *temp_keys;
node **temp_pointers;
temp_pointers = malloc((order + 1) * sizeof(node *));
if (temp_pointers == NULL) {
exit(EXIT_FAILURE);
}
temp_keys = malloc(order * sizeof(int));
if (temp_keys == NULL) {
exit(EXIT_FAILURE);
}
for (i = 0, j = 0; i < old_node->num_keys + 1; i++, j++) {
if (j == left_index + 1)
j++;
temp_pointers[j] = old_node->pointers[i];
}
for (i = 0, j = 0; i < old_node->num_keys; i++, j++) {
if (j == left_index)
j++;
temp_keys[j] = old_node->keys[i];
}
temp_pointers[left_index + 1] = right;
temp_keys[left_index] = key;
split = cut(order);
new_node = makeNode();
old_node->num_keys = 0;
for (i = 0; i < split - 1; i++) {
old_node->pointers[i] = temp_pointers[i];
old_node->keys[i] = temp_keys[i];
old_node->num_keys++;
}
old_node->pointers[i] = temp_pointers[i];
k_prime = temp_keys[split - 1];
for (++i, j = 0; i < order; i++, j++) {
new_node->pointers[j] = temp_pointers[i];
new_node->keys[j] = temp_keys[i];
new_node->num_keys++;
}
new_node->pointers[j] = temp_pointers[i];
free(temp_pointers);
free(temp_keys);
new_node->parent = old_node->parent;
for (i = 0; i <= new_node->num_keys; i++) {
child = new_node->pointers[i];
child->parent = new_node;
}
return insertIntoParent(root, old_node, k_prime, new_node);
}
node *insertIntoParent(node *root, node *left, int key, node *right){
int left_index;
node *parent;
parent = left->parent;
if (parent == NULL)
return insertIntoNewRoot(left, key, right);
left_index = getLeftIndex(parent, left);
if (parent->num_keys < order - 1)
return insertIntoNode(root, parent, left_index, key, right);
return insertIntoNodeAfterSplitting(root, parent, left_index, key, right);
}
node *insertIntoNewRoot(node *left, int key, node *right){
node *root = makeNode();
root->keys[0] = key;
root->pointers[0] = left;
root->pointers[1] = right;
root->num_keys++;
root->parent = NULL;
left->parent = root;
right->parent = root;
return root;
}
node *startNewTree(int key, record *pointer){
node *root = makeLeaf();
root->keys[0] = key;
root->pointers[0] = pointer;
root->pointers[order - 1] = NULL;
root->parent = NULL;
root->num_keys++;
return root;
}
node *insert(node *root, int key, int value){
record *record_pointer = NULL;
node *leaf = NULL;
record_pointer = find(root, key, false, NULL);
if (record_pointer != NULL) {
record_pointer->value = value;
return root;
}
record_pointer = makeRecord(value);
if (root == NULL)
return startNewTree(key, record_pointer);
leaf = findLeaf(root, key, false);
if (leaf->num_keys < order - 1) {
leaf = insertIntoLeaf(leaf, key, record_pointer);
return root;
}
return insertIntoLeafAfterSplitting(root, leaf, key, record_pointer);
}
int main(){
node *root;
char instruction;
root = NULL;
root = insert(root, 5, 33);
root = insert(root, 15, 21);
root = insert(root, 25, 31);
root = insert(root, 35, 41);
root = insert(root, 45, 10);
printTree(root);
findAndPrint(root, 15, instruction = 'a');
}Search Complexity
Time Complexity
If linear search is implemented inside a node, then total complexity is Θ(logt n).
If binary search is used, then total complexity is Θ(log2t.logt n).
B+ Tree Applications
- Multilevel Indexing
- Faster operations on the tree (insertion, deletion, search)
- Database indexing
Python Example for Beginners
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