Table of Contents
In this tutorial, you will learn how to insert a key into a btree. Also, you will find working examples of inserting keys into a B-tree in C, C++, Java and Python.
Inserting an element on a B-tree consists of two events: searching the appropriate node to insert the element and splitting the node if required.Insertion operation always takes place in the bottom-up approach.
Let us understand these events below.
1. Insertion Operation
- If the tree is empty, allocate a root node and insert the key.
- Update the allowed number of keys in the node.
- Search the appropriate node for insertion.
- If the node is full, follow the steps below.
- Insert the elements in increasing order.
- Now, there are elements greater than its limit. So, split at the median.
- Push the median key upwards and make the left keys as a left child and the right keys as a right child.
- If the node is not full, follow the steps below.
- Insert the node in increasing order.
2. Insertion Example
Let us understand the insertion operation with the illustrations below.
The elements to be inserted are 8, 9, 10, 11, 15, 16, 17, 18, 20, 23.

3. Algorithm for Inserting an Element
BreeInsertion(T, k) r root[T] if n[r] = 2t - 1 s = AllocateNode() root[T] = s leaf[s] = FALSE n[s] <- 0 c1[s] <- r BtreeSplitChild(s, 1, r) BtreeInsertNonFull(s, k) else BtreeInsertNonFull(r, k) BtreeInsertNonFull(x, k) i = n[x] if leaf[x] while i ≥ 1 and k < keyi[x] keyi+1 [x] = keyi[x] i = i - 1 keyi+1[x] = k n[x] = n[x] + 1 else while i ≥ 1 and k < keyi[x] i = i - 1 i = i + 1 if n[ci[x]] == 2t - 1 BtreeSplitChild(x, i, ci[x]) if k &rt; keyi[x] i = i + 1 BtreeInsertNonFull(ci[x], k) BtreeSplitChild(x, i) BtreeSplitChild(x, i, y) z = AllocateNode() leaf[z] = leaf[y] n[z] = t - 1 for j = 1 to t - 1 keyj[z] = keyj+t[y] if not leaf [y] for j = 1 to t cj[z] = cj + t[y] n[y] = t - 1 for j = n[x] + 1 to i + 1 cj+1[x] = cj[x] ci+1[x] = z for j = n[x] to i keyj+1[x] = keyj[x] keyi[x] = keyt[y] n[x] = n[x] + 1
4. Python, Java and C/C++ Examples
Source code by Python Language:
# Inserting a key on a B-tree in Python # Create a node class BTreeNode: def __init__(self, leaf=False): self.leaf = leaf self.keys = [] self.child = [] # Tree class BTree: def __init__(self, t): self.root = BTreeNode(True) self.t = t # Insert node def insert(self, k): root = self.root if len(root.keys) == (2 * self.t) - 1: temp = BTreeNode() self.root = temp temp.child.insert(0, root) self.split_child(temp, 0) self.insert_non_full(temp, k) else: self.insert_non_full(root, k) # Insert nonfull def insert_non_full(self, x, k): i = len(x.keys) - 1 if x.leaf: x.keys.append((None, None)) while i >= 0 and k[0] < x.keys[i][0]: x.keys[i + 1] = x.keys[i] i -= 1 x.keys[i + 1] = k else: while i >= 0 and k[0] < x.keys[i][0]: i -= 1 i += 1 if len(x.child[i].keys) == (2 * self.t) - 1: self.split_child(x, i) if k[0] > x.keys[i][0]: i += 1 self.insert_non_full(x.child[i], k) # Split the child def split_child(self, x, i): t = self.t y = x.child[i] z = BTreeNode(y.leaf) x.child.insert(i + 1, z) x.keys.insert(i, y.keys[t - 1]) z.keys = y.keys[t: (2 * t) - 1] y.keys = y.keys[0: t - 1] if not y.leaf: z.child = y.child[t: 2 * t] y.child = y.child[0: t - 1] # Print the tree def print_tree(self, x, l=0): print("Level ", l, " ", len(x.keys), end=":") for i in x.keys: print(i, end=" ") print() l += 1 if len(x.child) > 0: for i in x.child: self.print_tree(i, l) def main(): B = BTree(3) for i in range(10): B.insert((i, 2 * i)) B.print_tree(B.root) if __name__ == '__main__': main()
Source code by Java Language:
// Inserting a key on a B-tree in Java public class BTree { private int T; // Node Creation public class Node { int n; int key[] = new int[2 * T - 1]; Node child[] = new Node[2 * T]; boolean leaf = true; public int Find(int k) { for (int i = 0; i < this.n; i++) { if (this.key[i] == k) { return i; } } return -1; }; } public BTree(int t) { T = t; root = new Node(); root.n = 0; root.leaf = true; } private Node root; // split private void split(Node x, int pos, Node y) { Node z = new Node(); z.leaf = y.leaf; z.n = T - 1; for (int j = 0; j < T - 1; j++) { z.key[j] = y.key[j + T]; } if (!y.leaf) { for (int j = 0; j < T; j++) { z.child[j] = y.child[j + T]; } } y.n = T - 1; for (int j = x.n; j >= pos + 1; j--) { x.child[j + 1] = x.child[j]; } x.child[pos + 1] = z; for (int j = x.n - 1; j >= pos; j--) { x.key[j + 1] = x.key[j]; } x.key[pos] = y.key[T - 1]; x.n = x.n + 1; } // insert key public void insert(final int key) { Node r = root; if (r.n == 2 * T - 1) { Node s = new Node(); root = s; s.leaf = false; s.n = 0; s.child[0] = r; split(s, 0, r); _insert(s, key); } else { _insert(r, key); } } // insert node final private void _insert(Node x, int k) { if (x.leaf) { int i = 0; for (i = x.n - 1; i >= 0 && k < x.key[i]; i--) { x.key[i + 1] = x.key[i]; } x.key[i + 1] = k; x.n = x.n + 1; } else { int i = 0; for (i = x.n - 1; i >= 0 && k < x.key[i]; i--) { } ; i++; Node tmp = x.child[i]; if (tmp.n == 2 * T - 1) { split(x, i, tmp); if (k > x.key[i]) { i++; } } _insert(x.child[i], k); } } public void display() { display(root); } // Display the tree private void display(Node x) { assert (x == null); for (int i = 0; i < x.n; i++) { System.out.print(x.key[i] + " "); } if (!x.leaf) { for (int i = 0; i < x.n + 1; i++) { display(x.child[i]); } } } public static void main(String[] args) { BTree b = new BTree(3); b.insert(8); b.insert(9); b.insert(10); b.insert(11); b.insert(15); b.insert(20); b.insert(17); b.display(); } }
Source code by C Language:
// insertioning a key on a B-tree in C #include <stdio.h> #include <stdlib.h> #define MAX 3 #define MIN 2 struct btreeNode { int item[MAX + 1], count; struct btreeNode *link[MAX + 1]; }; struct btreeNode *root; // Node creation struct btreeNode *createNode(int item, struct btreeNode *child) { struct btreeNode *newNode; newNode = (struct btreeNode *)malloc(sizeof(struct btreeNode)); newNode->item[1] = item; newNode->count = 1; newNode->link[0] = root; newNode->link[1] = child; return newNode; } // Insert void insertValue(int item, int pos, struct btreeNode *node, struct btreeNode *child) { int j = node->count; while (j > pos) { node->item[j + 1] = node->item[j]; node->link[j + 1] = node->link[j]; j--; } node->item[j + 1] = item; node->link[j + 1] = child; node->count++; } // Split node void splitNode(int item, int *pval, int pos, struct btreeNode *node, struct btreeNode *child, struct btreeNode **newNode) { int median, j; if (pos > MIN) median = MIN + 1; else median = MIN; *newNode = (struct btreeNode *)malloc(sizeof(struct btreeNode)); j = median + 1; while (j <= MAX) { (*newNode)->item[j - median] = node->item[j]; (*newNode)->link[j - median] = node->link[j]; j++; } node->count = median; (*newNode)->count = MAX - median; if (pos <= MIN) { insertValue(item, pos, node, child); } else { insertValue(item, pos - median, *newNode, child); } *pval = node->item[node->count]; (*newNode)->link[0] = node->link[node->count]; node->count--; } // Set the value of node int setNodeValue(int item, int *pval, struct btreeNode *node, struct btreeNode **child) { int pos; if (!node) { *pval = item; *child = NULL; return 1; } if (item < node->item[1]) { pos = 0; } else { for (pos = node->count; (item < node->item[pos] && pos > 1); pos--) ; if (item == node->item[pos]) { printf("Duplicates not allowed\n"); return 0; } } if (setNodeValue(item, pval, node->link[pos], child)) { if (node->count < MAX) { insertValue(*pval, pos, node, *child); } else { splitNode(*pval, pval, pos, node, *child, child); return 1; } } return 0; } // Insert the value void insertion(int item) { int flag, i; struct btreeNode *child; flag = setNodeValue(item, &i, root, &child); if (flag) root = createNode(i, child); } // Copy the successor void copySuccessor(struct btreeNode *myNode, int pos) { struct btreeNode *dummy; dummy = myNode->link[pos]; for (; dummy->link[0] != NULL;) dummy = dummy->link[0]; myNode->item[pos] = dummy->item[1]; } // Do rightshift void rightShift(struct btreeNode *myNode, int pos) { struct btreeNode *x = myNode->link[pos]; int j = x->count; while (j > 0) { x->item[j + 1] = x->item[j]; x->link[j + 1] = x->link[j]; } x->item[1] = myNode->item[pos]; x->link[1] = x->link[0]; x->count++; x = myNode->link[pos - 1]; myNode->item[pos] = x->item[x->count]; myNode->link[pos] = x->link[x->count]; x->count--; return; } // Do leftshift void leftShift(struct btreeNode *myNode, int pos) { int j = 1; struct btreeNode *x = myNode->link[pos - 1]; x->count++; x->item[x->count] = myNode->item[pos]; x->link[x->count] = myNode->link[pos]->link[0]; x = myNode->link[pos]; myNode->item[pos] = x->item[1]; x->link[0] = x->link[1]; x->count--; while (j <= x->count) { x->item[j] = x->item[j + 1]; x->link[j] = x->link[j + 1]; j++; } return; } // Merge the nodes void mergeNodes(struct btreeNode *myNode, int pos) { int j = 1; struct btreeNode *x1 = myNode->link[pos], *x2 = myNode->link[pos - 1]; x2->count++; x2->item[x2->count] = myNode->item[pos]; x2->link[x2->count] = myNode->link[0]; while (j <= x1->count) { x2->count++; x2->item[x2->count] = x1->item[j]; x2->link[x2->count] = x1->link[j]; j++; } j = pos; while (j < myNode->count) { myNode->item[j] = myNode->item[j + 1]; myNode->link[j] = myNode->link[j + 1]; j++; } myNode->count--; free(x1); } // Adjust the node void adjustNode(struct btreeNode *myNode, int pos) { if (!pos) { if (myNode->link[1]->count > MIN) { leftShift(myNode, 1); } else { mergeNodes(myNode, 1); } } else { if (myNode->count != pos) { if (myNode->link[pos - 1]->count > MIN) { rightShift(myNode, pos); } else { if (myNode->link[pos + 1]->count > MIN) { leftShift(myNode, pos + 1); } else { mergeNodes(myNode, pos); } } } else { if (myNode->link[pos - 1]->count > MIN) rightShift(myNode, pos); else mergeNodes(myNode, pos); } } } // Traverse the tree void traversal(struct btreeNode *myNode) { int i; if (myNode) { for (i = 0; i < myNode->count; i++) { traversal(myNode->link[i]); printf("%d ", myNode->item[i + 1]); } traversal(myNode->link[i]); } } int main() { int item, ch; insertion(8); insertion(9); insertion(10); insertion(11); insertion(15); insertion(16); insertion(17); insertion(18); insertion(20); insertion(23); traversal(root); }
Source code by C++ Language:
// Inserting a key on a B-tree in C++ #include <iostream> using namespace std; class Node { int *keys; int t; Node **C; int n; bool leaf; public: Node(int _t, bool _leaf); void insertNonFull(int k); void splitChild(int i, Node *y); void traverse(); friend class BTree; }; class BTree { Node *root; int t; public: BTree(int _t) { root = NULL; t = _t; } void traverse() { if (root != NULL) root->traverse(); } void insert(int k); }; Node::Node(int t1, bool leaf1) { t = t1; leaf = leaf1; keys = new int[2 * t - 1]; C = new Node *[2 * t]; n = 0; } // Traverse the nodes void Node::traverse() { int i; for (i = 0; i < n; i++) { if (leaf == false) C[i]->traverse(); cout << " " << keys[i]; } if (leaf == false) C[i]->traverse(); } // Insert the node void BTree::insert(int k) { if (root == NULL) { root = new Node(t, true); root->keys[0] = k; root->n = 1; } else { if (root->n == 2 * t - 1) { Node *s = new Node(t, false); s->C[0] = root; s->splitChild(0, root); int i = 0; if (s->keys[0] < k) i++; s->C[i]->insertNonFull(k); root = s; } else root->insertNonFull(k); } } // Insert non full condition void Node::insertNonFull(int k) { int i = n - 1; if (leaf == true) { while (i >= 0 && keys[i] > k) { keys[i + 1] = keys[i]; i--; } keys[i + 1] = k; n = n + 1; } else { while (i >= 0 && keys[i] > k) i--; if (C[i + 1]->n == 2 * t - 1) { splitChild(i + 1, C[i + 1]); if (keys[i + 1] < k) i++; } C[i + 1]->insertNonFull(k); } } // split the child void Node::splitChild(int i, Node *y) { Node *z = new Node(y->t, y->leaf); z->n = t - 1; for (int j = 0; j < t - 1; j++) z->keys[j] = y->keys[j + t]; if (y->leaf == false) { for (int j = 0; j < t; j++) z->C[j] = y->C[j + t]; } y->n = t - 1; for (int j = n; j >= i + 1; j--) C[j + 1] = C[j]; C[i + 1] = z; for (int j = n - 1; j >= i; j--) keys[j + 1] = keys[j]; keys[i] = y->keys[t - 1]; n = n + 1; } int main() { BTree t(3); t.insert(8); t.insert(9); t.insert(10); t.insert(11); t.insert(15); t.insert(16); t.insert(17); t.insert(18); t.insert(20); t.insert(23); cout << "The B-tree is: "; t.traverse(); }