Prerequisite: Introduction of B+ trees
In this article, we will discuss that how to insert a node in B+ Tree. During insertion following properties of B+ Tree must be followed:
- Each node except root can have a maximum of M children and at least ceil(M/2) children.
- Each node can contain a maximum of M – 1 keys and a minimum of ceil(M/2) – 1 keys.
- The root has at least two children and atleast one search key.
- While insertion overflow of the node occurs when it contains more than M – 1 search key values.
Here M is the order of B+ tree.
Steps for insertion in B+ Tree
- Every element is inserted into a leaf node. So, go to the appropriate leaf node.
- Insert the key into the leaf node in increasing order only if there is no overflow. If there is an overflow go ahead with the following steps mentioned below to deal with overflow while maintaining the B+ Tree properties.
Properties for insertion B+ Tree
Case 1: Overflow in leaf node
- Split the leaf node into two nodes.
- First node contains ceil((m-1)/2) values.
- Second node contains the remaining values.
- Copy the smallest search key value from second node to the parent node.(Right biased)
Below is the illustration of inserting 8 into B+ Tree of order of 5:
Case 2: Overflow in non-leaf node
- Split the non leaf node into two nodes.
- First node contains ceil(m/2)-1 values.
- Move the smallest among remaining to the parent.
- Second node contains the remaining keys.
Below is the illustration of inserting 15 into B+ Tree of order of 5:
Example to illustrate insertion on a B+ tree
Problem: Insert the following key values 6, 16, 26, 36, 46 on a B+ tree with order = 3.
Solution:
Step 1: The order is 3 so at maximum in a node so there can be only 2 search key values. As insertion happens on a leaf node only in a B+ tree so insert search key value 6 and 16 in increasing order in the node. Below is the illustration of the same:
Step 2: We cannot insert 26 in the same node as it causes an overflow in the leaf node, We have to split the leaf node according to the rules. First part contains ceil((3-1)/2) values i.e., only 6. The second node contains the remaining values i.e., 16 and 26. Then also copy the smallest search key value from the second node to the parent node i.e., 16 to the parent node. Below is the illustration of the same:
Step 3: Now the next value is 36 that is to be inserted after 26 but in that node, it causes an overflow again in that leaf node. Again follow the above steps to split the node. First part contains ceil((3-1)/2) values i.e., only 16. The second node contains the remaining values i.e., 26 and 36. Then also copy the smallest search key value from the second node to the parent node i.e., 26 to the parent node. Below is the illustration of the same:
The illustration is shown in the diagram below.
Step 4: Now we have to insert 46 which is to be inserted after 36 but it causes an overflow in the leaf node. So we split the node according to the rules. The first part contains 26 and the second part contains 36 and 46 but now we also have to copy 36 to the parent node but it causes overflow as only two search key values can be accommodated in a node. Now follow the steps to deal with overflow in the non-leaf node.
First node contains ceil(3/2)-1 values i.e. ’16’.
Move the smallest among remaining to the parent i.e ’26’ will be the new parent node.
The second node contains the remaining keys i.e ’36’ and the rest of the leaf nodes remain the same. Below is the illustration of the same:
Below is the python implementation of B+ tree:
C++ #include <iostream> #include <vector> #include <cmath> #include <algorithm> using namespace std; // Node class class Node { public: int t; // Order of the B+ tree vector<string> keys; // Keys in the node vector<vector<int>> values; vector<Node*> child_ptr; // Pointers to child nodes bool leaf; // Boolean to check if the node is a leaf int n; // Current number of keys Node* ptr2next; // Pointer to the next node // Node constructor Node(int _t, Node* _ptr2next = NULL) { t = _t; ptr2next = _ptr2next; leaf = true; keys.resize(2*t-1); values.resize(2*t-1); child_ptr.resize(2*t); n = 0; } // Function to insert a key in a non-full node void insertNonFull(string k, int v) { int i = n-1; if (leaf) { keys.insert(keys.begin()+n, k); values.insert(values.begin()+n, vector<int>(1, v)); n += 1; while (i>=0 && keys[i]>k) { swap(keys[i], keys[i+1]); swap(values[i], values[i+1]); i -= 1; } } else { while (i>=0 && keys[i]>k) { i -= 1; } i += 1; if (child_ptr[i]->n == 2*t-1) { splitChild(i); if (keys[i] < k) { i += 1; } } child_ptr[i]->insertNonFull(k, v); } } // Function to split the child void splitChild(int i) { Node* y = child_ptr[i]; Node* z = new Node(y->t, y->ptr2next); child_ptr.insert(child_ptr.begin()+i+1, z); keys.insert(keys.begin()+i, y->keys[t-1]); values.insert(values.begin()+i, y->values[t-1]); y->ptr2next = z; z->leaf = y->leaf; z->n = t-1; y->n = t-1; for (int j=0; j<t-1; j++) { z->keys[j] = y->keys[j+t]; z->values[j] = y->values[j+t]; } if (!y->leaf) { for (int j=0; j<t; j++) { z->child_ptr[j] = y->child_ptr[j+t]; } } n += 1; } // Function to print the tree void print() { for (int i=0; i<n; i++) { if (!leaf) { child_ptr[i]->print(); } cout << "['" << keys[i] << "']" << endl; } if (!leaf) { child_ptr[n]->print(); } } // Function to search a key in the tree Node* search(string k, int v) { int i = 0; while (i<n && k>keys[i]) { i += 1; } if (keys[i] == k) { for (int j = 0; j < values[i].size(); j++) { if (values[i][j] == v) { return this; } } } if (leaf) { return NULL; } else { return child_ptr[i]->search(k, v); } } }; class BTree { public: Node* root; // Root of the B+ tree int t; // Order of the B+ tree // BTree constructor BTree(int _t) { root = new Node(_t); root->leaf = true; } // Function to insert a key in the tree void insert(string k, int v) { Node* r = root; if (r->n == 2*t-1) { Node* s = new Node(t); root = s; s->child_ptr[0] = r; s->splitChild(0); s->insertNonFull(k, v); } else { r->insertNonFull(k, v); } } // Function to print the tree void print() { root->print(); } // Function to search a key in the tree Node* search(string k, int v) { return (root == NULL)? NULL : root->search(k, v); } }; // Function to print the tree void printTree(BTree* tree) { tree->print(); } int main() { int record_len = 3; BTree* bplustree = new BTree(record_len); bplustree->insert("5", 33); bplustree->insert("15", 21); bplustree->insert("25", 31); bplustree->insert("35", 41); bplustree->insert("45", 10); printTree(bplustree); if (bplustree->search("5", 34) != NULL) { cout << "Found" << endl; } else { cout << "Not found" << endl; } return 0; } import java.util.ArrayList; import java.util.List; // Node creation class Node { int order; List<String> values; List<List<Node>> keys; Node nextKey; Node parent; boolean isLeaf; // Node constructor public Node(int order) { this.order = order; this.values = new ArrayList<>(); this.keys = new ArrayList<>(); this.nextKey = null; this.parent = null; this.isLeaf = false; } // Insert at the leaf public void insertAtLeaf(String value, Node key) { if (!this.values.isEmpty()) { for (int i = 0; i < this.values.size(); i++) { if (value.equals(this.values.get(i))) { this.keys.get(i).add(key); break; } else if (value.compareTo(this.values.get(i)) < 0) { this.values.add(i, value); this.keys.add(i, new ArrayList<>()); this.keys.get(i).add(key); break; } else if (i + 1 == this.values.size()) { this.values.add(value); this.keys.add(new ArrayList<>()); this.keys.get(i + 1).add(key); break; } } } else { this.values.add(value); this.keys.add(new ArrayList<>()); this.keys.get(0).add(key); } } } // B plus tree class BplusTree { Node root; // B plus tree constructor public BplusTree(int order) { this.root = new Node(order); this.root.isLeaf = true; } // Insert operation public void insert(String value, Node key) { Node oldNode = this.search(value); oldNode.insertAtLeaf(value, key); if (oldNode.values.size() == oldNode.order) { Node newNode = new Node(oldNode.order); newNode.isLeaf = true; newNode.parent = oldNode.parent; int mid = (int) Math.ceil(oldNode.order / 2.0) - 1; newNode.values = new ArrayList<>(oldNode.values.subList(mid + 1, oldNode.values.size())); newNode.keys = new ArrayList<>(oldNode.keys.subList(mid + 1, oldNode.keys.size())); newNode.nextKey = oldNode.nextKey; oldNode.values = new ArrayList<>(oldNode.values.subList(0, mid + 1)); oldNode.keys = new ArrayList<>(oldNode.keys.subList(0, mid + 1)); oldNode.nextKey = newNode; this.insertInParent(oldNode, newNode.values.get(0), newNode); } } // Search operation for different operations public Node search(String value) { Node currentNode = this.root; while (!currentNode.isLeaf) { for (int i = 0; i < currentNode.values.size(); i++) { if (value.equals(currentNode.values.get(i))) { currentNode = currentNode.keys.get(i + 1).get(0); break; } else if (value.compareTo(currentNode.values.get(i)) < 0) { currentNode = currentNode.keys.get(i).get(0); break; } else if (i + 1 == currentNode.values.size()) { currentNode = currentNode.keys.get(i + 1).get(0); break; } } } return currentNode; } // Find the node public boolean find(String value, Node key) { Node leaf = this.search(value); for (int i = 0; i < leaf.values.size(); i++) { if (leaf.values.get(i).equals(value)) { if (leaf.keys.get(i).contains(key)) { return true; } else { return false; } } } return false; } // Inserting at the parent public void insertInParent(Node n, String value, Node ndash) { if (this.root == n) { Node rootNode = new Node(n.order); rootNode.values.add(value); rootNode.keys.add(new ArrayList<>()); rootNode.keys.add(new ArrayList<>()); rootNode.keys.get(0).add(n); rootNode.keys.get(1).add(ndash); this.root = rootNode; n.parent = rootNode; ndash.parent = rootNode; return; } Node parentNode = n.parent; for (int i = 0; i < parentNode.keys.size(); i++) { if (parentNode.keys.get(i).get(0) == n) { parentNode.values.add(i, value); parentNode.keys.add(i + 1, new ArrayList<>()); parentNode.keys.get(i + 1).add(ndash); if (parentNode.keys.size() > parentNode.order) { Node parentdash = new Node(parentNode.order); parentdash.parent = parentNode.parent; int mid = (int) Math.ceil(parentNode.order / 2.0) - 1; parentdash.values = new ArrayList<>(parentNode.values.subList(mid + 1, parentNode.values.size())); parentdash.keys = new ArrayList<>(parentNode.keys.subList(mid + 1, parentNode.keys.size())); String value_ = parentNode.values.get(mid); if (mid == 0) { parentNode.values = new ArrayList<>(parentNode.values.subList(0, mid + 1)); } else { parentNode.values = new ArrayList<>(parentNode.values.subList(0, mid)); } parentNode.keys = new ArrayList<>(parentNode.keys.subList(0, mid + 1)); for (int j = 0; j < parentNode.keys.size(); j++) { parentNode.keys.get(j).get(0).parent = parentNode; } for (int j = 0; j < parentdash.keys.size(); j++) { parentdash.keys.get(j).get(0).parent = parentdash; } this.insertInParent(parentNode, value_, parentdash); } break; } } } } public class Main { public static void main(String[] args) { BplusTree bplusTree = new BplusTree(3); bplusTree.insert("5", new Node(3)); bplusTree.insert("15", new Node(3)); bplusTree.insert("25", new Node(3)); bplusTree.insert("35", new Node(3)); bplusTree.insert("45", new Node(3)); printTree(bplusTree); if (bplusTree.find("5", new Node(3))) { System.out.println("Found"); } else { System.out.println("Not found"); } } // Print the tree public static void printTree(BplusTree tree) { List<Node> lst = new ArrayList<>(); lst.add(tree.root); List<Integer> level = new ArrayList<>(); level.add(0); Node leaf = null; int flag = 0; int lev_leaf = 0; while (!lst.isEmpty()) { Node x = lst.remove(0); int lev = level.remove(0); if (!x.isLeaf) { for (int i = 0; i < x.keys.size(); i++) { System.out.println(x.keys.get(i).get(0).values); } } else { for (int i = 0; i < x.keys.size(); i++) { System.out.println(x.keys.get(i).get(0).values); } if (flag == 0) { lev_leaf = lev; leaf = x; flag = 1; } } } } } # Python3 program for implementing B+ Tree import math # Node creation class Node: def __init__(self, order): self.order = order self.values = [] self.keys = [] self.nextKey = None self.parent = None self.check_leaf = False # Insert at the leaf def insert_at_leaf(self, leaf, value, key): if (self.values): temp1 = self.values for i in range(len(temp1)): if (value == temp1[i]): self.keys[i].append(key) break elif (value < temp1[i]): self.values = self.values[:i] + [value] + self.values[i:] self.keys = self.keys[:i] + [[key]] + self.keys[i:] break elif (i + 1 == len(temp1)): self.values.append(value) self.keys.append([key]) break else: self.values = [value] self.keys = [[key]] # B plus tree class BplusTree: def __init__(self, order): self.root = Node(order) self.root.check_leaf = True # Insert operation def insert(self, value, key): value = str(value) old_node = self.search(value) old_node.insert_at_leaf(old_node, value, key) if (len(old_node.values) == old_node.order): node1 = Node(old_node.order) node1.check_leaf = True node1.parent = old_node.parent mid = int(math.ceil(old_node.order / 2)) - 1 node1.values = old_node.values[mid + 1:] node1.keys = old_node.keys[mid + 1:] node1.nextKey = old_node.nextKey old_node.values = old_node.values[:mid + 1] old_node.keys = old_node.keys[:mid + 1] old_node.nextKey = node1 self.insert_in_parent(old_node, node1.values[0], node1) # Search operation for different operations def search(self, value): current_node = self.root while(current_node.check_leaf == False): temp2 = current_node.values for i in range(len(temp2)): if (value == temp2[i]): current_node = current_node.keys[i + 1] break elif (value < temp2[i]): current_node = current_node.keys[i] break elif (i + 1 == len(current_node.values)): current_node = current_node.keys[i + 1] break return current_node # Find the node def find(self, value, key): l = self.search(value) for i, item in enumerate(l.values): if item == value: if key in l.keys[i]: return True else: return False return False # Inserting at the parent def insert_in_parent(self, n, value, ndash): if (self.root == n): rootNode = Node(n.order) rootNode.values = [value] rootNode.keys = [n, ndash] self.root = rootNode n.parent = rootNode ndash.parent = rootNode return parentNode = n.parent temp3 = parentNode.keys for i in range(len(temp3)): if (temp3[i] == n): parentNode.values = parentNode.values[:i] + \ [value] + parentNode.values[i:] parentNode.keys = parentNode.keys[:i + 1] + [ndash] + parentNode.keys[i + 1:] if (len(parentNode.keys) > parentNode.order): parentdash = Node(parentNode.order) parentdash.parent = parentNode.parent mid = int(math.ceil(parentNode.order / 2)) - 1 parentdash.values = parentNode.values[mid + 1:] parentdash.keys = parentNode.keys[mid + 1:] value_ = parentNode.values[mid] if (mid == 0): parentNode.values = parentNode.values[:mid + 1] else: parentNode.values = parentNode.values[:mid] parentNode.keys = parentNode.keys[:mid + 1] for j in parentNode.keys: j.parent = parentNode for j in parentdash.keys: j.parent = parentdash self.insert_in_parent(parentNode, value_, parentdash) # Print the tree def printTree(tree): lst = [tree.root] level = [0] leaf = None flag = 0 lev_leaf = 0 node1 = Node(str(level[0]) + str(tree.root.values)) while (len(lst) != 0): x = lst.pop(0) lev = level.pop(0) if (x.check_leaf == False): for i, item in enumerate(x.keys): print(item.values) else: for i, item in enumerate(x.keys): print(item.values) if (flag == 0): lev_leaf = lev leaf = x flag = 1 record_len = 3 bplustree = BplusTree(record_len) bplustree.insert('5', '33') bplustree.insert('15', '21') bplustree.insert('25', '31') bplustree.insert('35', '41') bplustree.insert('45', '10') printTree(bplustree) if(bplustree.find('5', '34')): print("Found") else: print("Not found") using System; using System.Collections.Generic; public class Node { public int t; // Order of the B+ tree public List<string> keys; // Keys in the node public List<List<int>> values; public List<Node> child_ptr; // Pointers to child nodes public bool leaf; // Boolean to check if the node is a leaf public int n; // Current number of keys public Node ptr2next; // Pointer to the next node // Node constructor public Node(int _t, Node _ptr2next = null) { t = _t; ptr2next = _ptr2next; leaf = true; keys = new List<string>(2 * t - 1); values = new List<List<int>>(2 * t - 1); child_ptr = new List<Node>(2 * t); n = 0; } // Function to insert a key in a non-full node public void InsertNonFull(string k, int v) { int i = n - 1; if (leaf) { keys.Insert(n, k); values.Insert(n, new List<int> { v }); n += 1; while (i >= 0 && string.Compare(keys[i], k) > 0) { string tempKey = keys[i]; keys[i] = keys[i + 1]; keys[i + 1] = tempKey; List<int> tempValue = values[i]; values[i] = values[i + 1]; values[i + 1] = tempValue; i -= 1; } } else { while (i >= 0 && string.Compare(keys[i], k) > 0) { i -= 1; } i += 1; if (child_ptr[i].n == 2 * t - 1) { SplitChild(i); if (string.Compare(keys[i], k) < 0) { i += 1; } } child_ptr[i].InsertNonFull(k, v); } } // Function to split the child public void SplitChild(int i) { Node y = child_ptr[i]; Node z = new Node(y.t, y.ptr2next); child_ptr.Insert(i + 1, z); keys.Insert(i, y.keys[t - 1]); values.Insert(i, y.values[t - 1]); y.ptr2next = z; z.leaf = y.leaf; z.n = t - 1; y.n = t - 1; for (int j = 0; j < t - 1; j++) { z.keys[j] = y.keys[j + t]; z.values[j] = y.values[j + t]; } if (!y.leaf) { for (int j = 0; j < t; j++) { z.child_ptr[j] = y.child_ptr[j + t]; } } n += 1; } // Function to print the tree public void Print() { for (int i = 0; i < n; i++) { if (!leaf) { child_ptr[i].Print(); } Console.WriteLine("['" + keys[i] + "']"); } if (!leaf) { child_ptr[n].Print(); } } // Function to search a key in the tree public Node Search(string k, int v) { int i = 0; while (i < n && string.Compare(keys[i], k) < 0) { i += 1; } if (keys[i] == k) { for (int j = 0; j < values[i].Count; j++) { if (values[i][j] == v) { return this; } } } if (leaf) { return null; } else { return child_ptr[i].Search(k, v); } } } public class BTree { public Node root; // Root of the B+ tree public int t; // Order of the B+ tree // BTree constructor public BTree(int _t) { root = new Node(_t); root.leaf = true; t = _t; } // Function to insert a key in the tree public void Insert(string k, int v) { Node r = root; if (r.n == 2 * t - 1) { Node s = new Node(t); root = s; s.child_ptr.Add(r); s.SplitChild(0); s.InsertNonFull(k, v); } else { r.InsertNonFull(k, v); } } // Function to print the tree public void Print() { root.Print(); } // Function to search a key in the tree public Node Search(string k, int v) { return (root == null) ? null : root.Search(k, v); } } public class Program { // Function to print the tree public static void PrintTree(BTree tree) { tree.Print(); } public static void Main() { int record_len = 3; BTree bplustree = new BTree(record_len); bplustree.Insert("5", 33); bplustree.Insert("15", 21); bplustree.Insert("25", 31); bplustree.Insert("35", 41); bplustree.Insert("45", 10); PrintTree(bplustree); if (bplustree.Search("5", 34) != null) { Console.WriteLine("Found"); } else { Console.WriteLine("Not found"); } } } // JavaScript equivalent // Node creation class Node { constructor(order) { this.order = order; this.values = []; this.keys = []; this.nextKey = null; this.parent = null; this.check_leaf = false; } // Insert at the leaf insert_at_leaf(leaf, value, key) { if (this.values.length !== 0) { const temp1 = this.values; for (let i = 0; i < temp1.length; i++) { if (value == temp1[i]) { this.keys[i].push(key); break; } else if (value < temp1[i]) { this.values = [ ...this.values.slice(0, i), value, ...this.values.slice(i) ]; this.keys = [ ...this.keys.slice(0, i), [key], ...this.keys.slice(i) ]; break; } else if (i + 1 == temp1.length) { this.values.push(value); this.keys.push([key]); break; } } } else { this.values = [value]; this.keys = [[key]]; } } } // B plus tree class BplusTree { constructor(order) { this.root = new Node(order); this.root.check_leaf = true; } // Insert operation insert(value, key) { value = String(value); const old_node = this.search(value); old_node.insert_at_leaf(old_node, value, key); if (old_node.values.length == old_node.order) { const node1 = new Node(old_node.order); node1.check_leaf = true; node1.parent = old_node.parent; const mid = Math.ceil(old_node.order / 2) - 1; node1.values = old_node.values.slice(mid + 1); node1.keys = old_node.keys.slice(mid + 1); node1.nextKey = old_node.nextKey; old_node.values = old_node.values.slice(0, mid + 1); old_node.keys = old_node.keys.slice(0, mid + 1); old_node.nextKey = node1; this.insert_in_parent(old_node, node1.values[0], node1); } } // Search operation for different operations search(value) { let current_node = this.root; while (current_node.check_leaf == false) { const temp2 = current_node.values; for (let i = 0; i < temp2.length; i++) { if (value == temp2[i]) { current_node = current_node.keys[i + 1]; break; } else if (value < temp2[i]) { current_node = current_node.keys[i]; break; } else if (i + 1 == current_node.values.length) { current_node = current_node.keys[i + 1]; break; } } } return current_node; } // Find the node find(value, key) { const l = this.search(value); for (let i = 0; i < l.values.length; i++) { if (l.values[i] == value) { if (l.keys[i].includes(key)) { return true; } else { return false; } } } return false; } // Inserting at the parent insert_in_parent(n, value, ndash) { if (this.root == n) { const rootNode = new Node(n.order); rootNode.values = [value]; rootNode.keys = [n, ndash]; this.root = rootNode; n.parent = rootNode; ndash.parent = rootNode; return; } const parentNode = n.parent; const temp3 = parentNode.keys; for (let i = 0; i < temp3.length; i++) { if (temp3[i] == n) { parentNode.values = [ ...parentNode.values.slice(0, i), value, ...parentNode.values.slice(i) ]; parentNode.keys = [ ...parentNode.keys.slice(0, i + 1), ndash, ...parentNode.keys.slice(i + 1) ]; if (parentNode.keys.length > parentNode.order) { const parentdash = new Node(parentNode.order); parentdash.parent = parentNode.parent; const mid = Math.ceil(parentNode.order / 2) - 1; parentdash.values = parentNode.values.slice(mid + 1); parentdash.keys = parentNode.keys.slice(mid + 1); const value_ = parentNode.values[mid]; if (mid == 0) { parentNode.values = parentNode.values.slice(0, mid + 1); } else { parentNode.values = parentNode.values.slice(0, mid); } parentNode.keys = parentNode.keys.slice(0, mid + 1); for (let j = 0; j < parentNode.keys.length; j++) { parentNode.keys[j].parent = parentNode; } for (let j = 0; j < parentdash.keys.length; j++) { parentdash.keys[j].parent = parentdash; } this.insert_in_parent(parentNode, value_, parentdash); } } } } } // Print the tree function printTree(tree) { let lst = [tree.root]; let level = [0]; let leaf = null; let flag = 0; let lev_leaf = 0; const node1 = new Node(String(level[0]) + String(tree.root.values)); while (lst.length !== 0) { const x = lst.shift(); const lev = level.shift(); if (x.check_leaf == false) { for (let i = 0; i < x.keys.length; i++) { console.log(x.keys[i].values); } } else { for (let i = 0; i < x.keys.length; i++) { console.log(x.keys[i].values); } if (flag == 0) { lev_leaf = lev; leaf = x; flag = 1; } } } } const record_len = 3; const bplustree = new BplusTree(record_len); bplustree.insert("5", "33"); bplustree.insert("15", "21"); bplustree.insert("25", "31"); bplustree.insert("35", "41"); bplustree.insert("45", "10"); printTree(bplustree); if (bplustree.find("5", "34")) { console.log("Found"); } else { console.log("Not found"); } Output['15'] ['25'] ['35'] ['45'] ['5'] Not found
Time complexity: O(log n)
Auxiliary Space: O(log n)
Below is the C++ implementation B+ tree:
C++ #include <bits/stdc++.h> using namespace std; typedef long long ll; typedef unsigned long long ull; #define pb push_back int bucketSize = 3; // Create 2 classes, one for node and one for tree; class node { public: bool isLeaf; node** ptr; int *key, size; node(); }; node::node() { key = new int[bucketSize]; ptr = new node*[bucketSize + 1](); } class Btree { public: // Root of tree stored here; node* root; Btree(); void deleteNode(int); int search(int); void display(node*); void insert(int); node* findParent(node*, node*); node* getRoot(); void shiftLevel(int, node*, node*); }; node* Btree::getRoot() { return root; } Btree::Btree() { root = NULL; } void Btree::insert(int x) { if (root == NULL) { root = new node; root->key[0] = x; root->isLeaf = true; root->size = 1; } else { node* current = root; node* parent; while (current->isLeaf == false) { parent = current; for (int i = 0; i < current->size; i++) { if (x < current->key[i]) { current = current->ptr[i]; break; } if (i == current->size - 1) { current = current->ptr[i + 1]; break; } } } // now we have reached leaf; if (current->size < bucketSize) { // if the node to be inserted is // not filled int i = 0; // Traverse btree while (x > current->key[i] && i < current->size) // goto pt where needs to be inserted. i++; for (int j = current->size; j > i; j--) // adjust and insert element; current->key[j] = current->key[j - 1]; current->key[i] = x; // size should be increased by 1 current->size++; current->ptr[current->size] = current->ptr[current->size - 1]; current->ptr[current->size - 1] = NULL; } // if block does not have enough space; else { node* newLeaf = new node; int tempNode[bucketSize + 1]; for (int i = 0; i < bucketSize; i++) // all elements of this block stored tempNode[i] = current->key[i]; int i = 0, j; // find the right posn of num to be inserted while (x > tempNode[i] && i < bucketSize) i++; for (int j = bucketSize + 1; j > i; j--) tempNode[j] = tempNode[j - 1]; tempNode[i] = x; // inserted element in its rightful position; newLeaf->isLeaf = true; current->size = (bucketSize + 1) / 2; newLeaf->size = (bucketSize + 1) - (bucketSize + 1) / 2; // now rearrangement begins! current->ptr[current->size] = newLeaf; newLeaf->ptr[newLeaf->size] = current->ptr[bucketSize]; current->ptr[newLeaf->size] = current->ptr[bucketSize]; current->ptr[bucketSize] = NULL; for (int i = 0; i < current->size; i++) current->key[i] = tempNode[i]; for (int i = 0, j = current->size; i < newLeaf->size; i++, j++) newLeaf->key[i] = tempNode[j]; // if this is root, then fine, // else we need to increase the height of tree; if (current == root) { node* newRoot = new node; newRoot->key[0] = newLeaf->key[0]; newRoot->ptr[0] = current; newRoot->ptr[1] = newLeaf; newRoot->isLeaf = false; newRoot->size = 1; root = newRoot; } else shiftLevel( newLeaf->key[0], parent, newLeaf); // parent->original root } } } void Btree::shiftLevel(int x, node* current, node* child) { // insert or create an internal node; if (current->size < bucketSize) { // if can fit in this level, do that int i = 0; while (x > current->key[i] && i < current->size) i++; for (int j = current->size; j > i; j--) current->key[j] = current->key[j - 1]; for (int j = current->size + 1; j > i + 1; j--) current->ptr[j] = current->ptr[j - 1]; current->key[i] = x; current->size++; current->ptr[i + 1] = child; } // shift up else { node* newInternal = new node; int tempKey[bucketSize + 1]; node* tempPtr[bucketSize + 2]; for (int i = 0; i < bucketSize; i++) tempKey[i] = current->key[i]; for (int i = 0; i < bucketSize + 1; i++) tempPtr[i] = current->ptr[i]; int i = 0, j; while (x > tempKey[i] && i < bucketSize) i++; for (int j = bucketSize + 1; j > i; j--) tempKey[j] = tempKey[j - 1]; tempKey[i] = x; for (int j = bucketSize + 2; j > i + 1; j--) tempPtr[j] = tempPtr[j - 1]; tempPtr[i + 1] = child; newInternal->isLeaf = false; current->size = (bucketSize + 1) / 2; newInternal->size = bucketSize - (bucketSize + 1) / 2; for (int i = 0, j = current->size + 1; i < newInternal->size; i++, j++) newInternal->key[i] = tempKey[j]; for (int i = 0, j = current->size + 1; i < newInternal->size + 1; i++, j++) newInternal->ptr[i] = tempPtr[j]; if (current == root) { node* newRoot = new node; newRoot->key[0] = current->key[current->size]; newRoot->ptr[0] = current; newRoot->ptr[1] = newInternal; newRoot->isLeaf = false; newRoot->size = 1; root = newRoot; } else shiftLevel(current->key[current->size], findParent(root, current), newInternal); } } int Btree::search(int x) { if (root == NULL) return -1; else { node* current = root; while (current->isLeaf == false) { for (int i = 0; i < current->size; i++) { if (x < current->key[i]) { current = current->ptr[i]; break; } if (i == current->size - 1) { current = current->ptr[i + 1]; break; } } } for (int i = 0; i < current->size; i++) { if (current->key[i] == x) { // cout<<"Key found "<<endl; return 1; // return; } } // cout<<"Key not found"<<endl; return 0; } } // Print the tree void Btree::display(node* current) { if (current == NULL) return; queue<node*> q; q.push(current); while (!q.empty()) { int l; l = q.size(); for (int i = 0; i < l; i++) { node* tNode = q.front(); q.pop(); for (int j = 0; j < tNode->size; j++) if (tNode != NULL) cout << tNode->key[j] << " "; for (int j = 0; j < tNode->size + 1; j++) if (tNode->ptr[j] != NULL) q.push(tNode->ptr[j]); cout << "\t"; } cout << endl; } } node* Btree::findParent(node* current, node* child) { node* parent; if (current->isLeaf || (current->ptr[0])->isLeaf) return NULL; for (int i = 0; i < current->size + 1; i++) { if (current->ptr[i] == child) { parent = current; return parent; } else { parent = findParent(current->ptr[i], child); if (parent != NULL) return parent; } } return parent; } signed main() { ios_base::sync_with_stdio(false); Btree node; cout << "The size of bucket is " << bucketSize << "! " << endl; node.insert(1); node.insert(2); node.insert(3); node.display(node.getRoot()); node.insert(4); node.insert(5); node.display(node.getRoot()); return 0; } import java.util.LinkedList; import java.util.Queue; // Node class represents a node in the B-tree class Node { boolean isLeaf; // Flag to check if the node is a leaf Node[] ptr; // Array of child pointers int[] key; // Array of keys int size; // Current number of keys in the node Node() { key = new int[BTree.bucketSize]; ptr = new Node[BTree.bucketSize + 1]; } } // BTree class represents the B-tree structure class BTree { static int bucketSize = 3; // Size of the bucket or order of the B-tree Node root; // Root of the B-tree BTree() { root = null; // Initialize the B-tree with no root initially } // Method to perform deletion in the B-tree (not implemented) void deleteNode(int x) { // Implement deletion logic if needed } // Method to search for a key in the B-tree int search(int x) { if (root == null) return -1; else { Node current = root; while (!current.isLeaf) { for (int i = 0; i < current.size; i++) { if (x < current.key[i]) { current = current.ptr[i]; break; } if (i == current.size - 1) { current = current.ptr[i + 1]; break; } } } for (int i = 0; i < current.size; i++) { if (current.key[i] == x) { // System.out.println("Key found "); return 1; } } // System.out.println("Key not found"); return 0; } } // Method to display the B-tree level by level using a queue void display(Node current) { if (current == null) return; Queue<Node> queue = new LinkedList<>(); queue.add(current); while (!queue.isEmpty()) { int l = queue.size(); for (int i = 0; i < l; i++) { Node tNode = queue.poll(); if (tNode != null) { for (int j = 0; j < tNode.size; j++) System.out.print(tNode.key[j] + " "); for (int j = 0; j < tNode.size + 1; j++) if (tNode.ptr[j] != null) queue.add(tNode.ptr[j]); System.out.print("\t"); } } System.out.println(); } } // Method to find the parent of a given node in the B-tree Node findParent(Node current, Node child) { Node parent = null; if (current.isLeaf || current.ptr[0].isLeaf) return null; for (int i = 0; i < current.size + 1; i++) { if (current.ptr[i] == child) { parent = current; return parent; } else { parent = findParent(current.ptr[i], child); if (parent != null) return parent; } } return parent; } // Method to insert a key into the B-tree void insert(int x) { if (root == null) { root = new Node(); root.key[0] = x; root.isLeaf = true; root.size = 1; } else { Node current = root; Node parent = null; while (!current.isLeaf) { parent = current; for (int i = 0; i < current.size; i++) { if (x < current.key[i]) { current = current.ptr[i]; break; } if (i == current.size - 1) { current = current.ptr[i + 1]; break; } } } if (current.size < bucketSize) { int i = 0; while (i < current.size && x > current.key[i]) { i++; } for (int j = current.size; j > i; j--) { current.key[j] = current.key[j - 1]; } current.key[i] = x; current.size++; current.ptr[current.size] = current.ptr[current.size - 1]; current.ptr[current.size - 1] = null; } else { Node newLeaf = new Node(); int[] tempNode = new int[bucketSize + 1]; for (int i = 0; i < bucketSize; i++) { tempNode[i] = current.key[i]; } int i = 0, j; while (i < bucketSize && x > tempNode[i]) { i++; } for (j = bucketSize; j > i; j--) { tempNode[j] = tempNode[j - 1]; } tempNode[i] = x; newLeaf.isLeaf = true; current.size = (bucketSize + 1) / 2; newLeaf.size = (bucketSize + 1) - (bucketSize + 1) / 2; current.ptr[current.size] = newLeaf; newLeaf.ptr[newLeaf.size] = current.ptr[bucketSize]; current.ptr[newLeaf.size] = current.ptr[bucketSize]; current.ptr[bucketSize] = null; for (i = 0; i < current.size; i++) { current.key[i] = tempNode[i]; } for (i = 0, j = current.size; i < newLeaf.size; i++, j++) { newLeaf.key[i] = tempNode[j]; } if (current == root) { Node newRoot = new Node(); newRoot.key[0] = newLeaf.key[0]; newRoot.ptr[0] = current; newRoot.ptr[1] = newLeaf; newRoot.isLeaf = false; newRoot.size = 1; root = newRoot; } else { shiftLevel(newLeaf.key[0], findParent(root, current), newLeaf); } } } } // Method to shift a level in the B-tree during insertion void shiftLevel(int x, Node current, Node child) { if (current.size < bucketSize) { int i = 0; while (x > current.key[i] && i < current.size) i++; for (int j = current.size; j > i; j--) current.key[j] = current.key[j - 1]; for (int j = current.size + 1; j > i + 1; j--) current.ptr[j] = current.ptr[j - 1]; current.key[i] = x; current.size++; current.ptr[i + 1] = child; } else { Node newInternal = new Node(); int[] tempKey = new int[bucketSize + 1]; Node[] tempPtr = new Node[bucketSize + 2]; for (int i = 0; i < bucketSize; i++) tempKey[i] = current.key[i]; for (int i = 0; i < bucketSize + 1; i++) tempPtr[i] = current.ptr[i]; int i = 0, j; while (x > tempKey[i] && i < bucketSize) i++; for (j = bucketSize + 1; j > i; j--) tempKey[j] = tempKey[j - 1]; tempKey[i] = x; for (j = bucketSize + 2; j > i + 1; j--) tempPtr[j] = tempPtr[j - 1]; tempPtr[i + 1] = child; newInternal.isLeaf = false; current.size = (bucketSize + 1) / 2; newInternal.size = bucketSize - (bucketSize + 1) / 2; for (i = 0, j = current.size + 1; i < newInternal.size; i++, j++) newInternal.key[i] = tempKey[j]; for (i = 0, j = current.size + 1; i < newInternal.size + 1; i++, j++) newInternal.ptr[i] = tempPtr[j]; if (current == root) { Node newRoot = new Node(); newRoot.key[0] = current.key[current.size]; newRoot.ptr[0] = current; newRoot.ptr[1] = newInternal; newRoot.isLeaf = false; newRoot.size = 1; root = newRoot; } else shiftLevel(current.key[current.size], findParent(root, current), newInternal); } } } // Main class for testing the B-tree implementation public class Main { public static void main(String[] args) { BTree bTree = new BTree(); System.out.println("The size of bucket is " + BTree.bucketSize + "! "); // Insert some keys into the B-tree bTree.insert(1); bTree.insert(2); bTree.insert(3); bTree.display(bTree.root); // Insert more keys and display the updated B-tree bTree.insert(4); bTree.insert(5); bTree.display(bTree.root); } } class Node: def __init__(self): self.is_leaf = False self.ptr = [None] * (bucket_size + 1) # Pointers to child nodes self.key = [None] * bucket_size # Keys stored in the node self.size = 0 # Number of keys currently in the node class BTree: def __init__(self): self.root = None # Root node of the B-tree def get_root(self): return self.root # Getter for the root node def insert(self, x): if self.root is None: # B-tree is empty, create a new root self.root = Node() self.root.key[0] = x self.root.is_leaf = True self.root.size = 1 else: current = self.root parent = None # Traverse the tree to find the appropriate leaf node for insertion while not current.is_leaf: parent = current for i in range(current.size): if current.key[i] is None or x < current.key[i]: current = current.ptr[i] break if i == current.size - 1: current = current.ptr[i + 1] break if current.size < bucket_size: # Insert into a non-full leaf node i = 0 while i < current.size and (current.key[i] is not None and x > current.key[i]): i += 1 # Shift keys and pointers to make space for the new key for j in range(current.size, i, -1): current.key[j] = current.key[j - 1] current.key[i] = x current.size += 1 current.ptr[current.size] = current.ptr[current.size - 1] current.ptr[current.size - 1] = None else: # Split the leaf node if it is full new_leaf = Node() temp_node = [None] * (bucket_size + 1) # Copy keys to temporary array for i in range(bucket_size): temp_node[i] = current.key[i] i = 0 while i < bucket_size and (temp_node[i] is not None and x > temp_node[i]): i += 1 # Shift keys in the temporary array to make space for the new key for j in range(bucket_size, i, -1): temp_node[j] = temp_node[j - 1] temp_node[i] = x # Update sizes of the current and new_leaf nodes new_leaf.is_leaf = True current.size = (bucket_size + 1) // 2 new_leaf.size = bucket_size + 1 - (bucket_size + 1) // 2 # Update pointers current.ptr[current.size] = new_leaf new_leaf.ptr[new_leaf.size] = current.ptr[bucket_size] current.ptr[new_leaf.size] = current.ptr[bucket_size] current.ptr[bucket_size] = None # Copy keys from the temporary array to the current and new_leaf nodes for i in range(current.size): current.key[i] = temp_node[i] for i in range(current.size, bucket_size): new_leaf.key[i - current.size] = temp_node[i] if current == self.root: # Update the root if splitting the root new_root = Node() new_root.key[0] = new_leaf.key[0] new_root.ptr[0] = current new_root.ptr[1] = new_leaf new_root.is_leaf = False new_root.size = 1 self.root = new_root else: # Propagate the split upwards self.shift_level(new_leaf.key[0], parent, new_leaf) def shift_level(self, x, current, child): # Helper method to handle splitting of non-leaf nodes if current.size < bucket_size: i = 0 while i < current.size and (current.key[i] is not None and x > current.key[i]): i += 1 # Shift keys and pointers to make space for the new key and child for j in range(current.size, i, -1): current.key[j] = current.key[j - 1] for j in range(current.size + 1, i + 1, -1): current.ptr[j] = current.ptr[j - 1] current.key[i] = x current.size += 1 current.ptr[i + 1] = child else: # Split the non-leaf node if it is full new_internal = Node() temp_key = [None] * (bucket_size + 1) temp_ptr = [None] * (bucket_size + 2) # Copy keys and pointers to temporary arrays for i in range(bucket_size): temp_key[i] = current.key[i] for i in range(bucket_size + 1): temp_ptr[i] = current.ptr[i] i = 0 while i < bucket_size and (temp_key[i] is not None and x > temp_key[i]): i += 1 # Shift keys in the temporary array to make space for the new key for j in range(bucket_size, i, -1): temp_key[j] = temp_key[j - 1] temp_key[i] = x # Shift pointers in the temporary array to make space for the new child for j in range(bucket_size + 1, i + 1, -1): temp_ptr[j] = temp_ptr[j - 1] temp_ptr[i + 1] = child new_internal.is_leaf = False current.size = (bucket_size + 1) // 2 new_internal.size = bucket_size - (bucket_size + 1) // 2 # Copy keys and pointers from the temporary arrays to the current and new_internal nodes for i in range(current.size + 1, bucket_size + 1): new_internal.key[i - current.size - 1] = temp_key[i] for i in range(current.size + 1, bucket_size + 2): new_internal.ptr[i - current.size - 1] = temp_ptr[i] if current == self.root: # Update the root if splitting the root new_root = Node() new_root.key[0] = current.key[current.size] new_root.ptr[0] = current new_root.ptr[1] = new_internal new_root.is_leaf = False new_root.size = 1 self.root = new_root else: # Propagate the split upwards self.shift_level(current.key[current.size], self.find_parent(self.root, current), new_internal) def search(self, x): # Search for a key in the B-tree if self.root is None: return -1 # B-tree is empty else: current = self.root while not current.is_leaf: for i in range(current.size): if x < current.key[i]: current = current.ptr[i] break if i == current.size - 1: current = current.ptr[i + 1] break for i in range(current.size): if current.key[i] == x: return 1 # Key found return 0 # Key not found def display(self, current): # Display the B-tree if current is None: return q = [current] while q: l = len(q) for _ in range(l): t_node = q.pop(0) for j in range(t_node.size): if t_node.key[j] is not None: print(t_node.key[j], end=' ') for j in range(t_node.size + 1): if t_node.ptr[j]: q.append(t_node.ptr[j]) print('\t', end='') print('\n') def find_parent(self, current, child): # Helper method to find the parent of a given node parent = None if current.is_leaf or current.ptr[0].is_leaf: return None # No parent for leaf nodes for i in range(current.size + 1): if current.ptr[i] == child: parent = current return parent else: parent = self.find_parent(current.ptr[i], child) if parent: return parent return parent bucket_size = 3 # Set the bucket size for the B-tree btree = BTree() # Create a new B-tree print('The size of bucket is', bucket_size, '!') # Insert elements into the B-tree btree.insert(1) btree.insert(2) btree.insert(3) btree.display(btree.get_root()) btree.insert(4) btree.insert(5) btree.display(btree.get_root()) class Node { constructor() { this.isLeaf = false; this.ptr = new Array(bucketSize + 1).fill(null); // Pointers to child nodes this.key = new Array(bucketSize); // Keys stored in the node this.size = 0; // Number of keys currently in the node } } class BTree { constructor() { this.root = null; // Root node of the B-tree } getRoot() { return this.root; // Getter for the root node } insert(x) { if (this.root === null) { // B-tree is empty, create a new root this.root = new Node(); this.root.key[0] = x; this.root.isLeaf = true; this.root.size = 1; } else { let current = this.root; let parent; // Traverse the tree to find the appropriate leaf node for insertion while (current.isLeaf === false) { parent = current; for (let i = 0; i < current.size; i++) { if (x < current.key[i]) { current = current.ptr[i]; break; } if (i === current.size - 1) { current = current.ptr[i + 1]; break; } } } if (current.size < bucketSize) { // Insert into a non-full leaf node let i = 0; while (x > current.key[i] && i < current.size) { i++; } // Shift keys and pointers to make space for the new key for (let j = current.size; j > i; j--) { current.key[j] = current.key[j - 1]; } current.key[i] = x; current.size++; current.ptr[current.size] = current.ptr[current.size - 1]; current.ptr[current.size - 1] = null; } else { // Split the leaf node if it is full let newLeaf = new Node(); let tempNode = new Array(bucketSize + 1); // Copy keys to temporary array for (let i = 0; i < bucketSize; i++) { tempNode[i] = current.key[i]; } let i = 0; while (x > tempNode[i] && i < bucketSize) { i++; } // Shift keys in the temporary array to make space for the new key for (let j = bucketSize + 1; j > i; j--) { tempNode[j] = tempNode[j - 1]; } tempNode[i] = x; // Update sizes of the current and newLeaf nodes newLeaf.isLeaf = true; current.size = Math.floor((bucketSize + 1) / 2); newLeaf.size = bucketSize + 1 - Math.floor((bucketSize + 1) / 2); // Update pointers current.ptr[current.size] = newLeaf; newLeaf.ptr[newLeaf.size] = current.ptr[bucketSize]; current.ptr[newLeaf.size] = current.ptr[bucketSize]; current.ptr[bucketSize] = null; // Copy keys from the temporary array to the current and newLeaf nodes for (let i = 0; i < current.size; i++) { current.key[i] = tempNode[i]; } for (let i = 0, j = current.size; i < newLeaf.size; i++, j++) { newLeaf.key[i] = tempNode[j]; } if (current === this.root) { // Update the root if splitting the root let newRoot = new Node(); newRoot.key[0] = newLeaf.key[0]; newRoot.ptr[0] = current; newRoot.ptr[1] = newLeaf; newRoot.isLeaf = false; newRoot.size = 1; this.root = newRoot; } else { // Propagate the split upwards this.shiftLevel(newLeaf.key[0], parent, newLeaf); } } } } shiftLevel(x, current, child) { // Helper method to handle splitting of non-leaf nodes if (current.size < bucketSize) { let i = 0; while (x > current.key[i] && i < current.size) { i++; } // Shift keys and pointers to make space for the new key and child for (let j = current.size; j > i; j--) { current.key[j] = current.key[j - 1]; } for (let j = current.size + 1; j > i + 1; j--) { current.ptr[j] = current.ptr[j - 1]; } current.key[i] = x; current.size++; current.ptr[i + 1] = child; } else { // Split the non-leaf node if it is full let newInternal = new Node(); let tempKey = new Array(bucketSize + 1); let tempPtr = new Array(bucketSize + 2); // Copy keys and pointers to temporary arrays for (let i = 0; i < bucketSize; i++) { tempKey[i] = current.key[i]; } for (let i = 0; i < bucketSize + 1; i++) { tempPtr[i] = current.ptr[i]; } let i = 0; while (x > tempKey[i] && i < bucketSize) { i++; } // Shift keys in the temporary array to make space for the new key for (let j = bucketSize + 1; j > i; j--) { tempKey[j] = tempKey[j - 1]; } tempKey[i] = x; // Shift pointers in the temporary array to make space for the new child for (let j = bucketSize + 2; j > i + 1; j--) { tempPtr[j] = tempPtr[j - 1]; } tempPtr[i + 1] = child; newInternal.isLeaf = false; current.size = Math.floor((bucketSize + 1) / 2); newInternal.size = bucketSize - Math.floor((bucketSize + 1) / 2); // Copy keys and pointers from the temporary arrays to the current and newInternal nodes for (let i = 0, j = current.size + 1; i < newInternal.size; i++, j++) { newInternal.key[i] = tempKey[j]; } for (let i = 0, j = current.size + 1; i < newInternal.size + 1; i++, j++) { newInternal.ptr[i] = tempPtr[j]; } if (current === this.root) { // Update the root if splitting the root let newRoot = new Node(); newRoot.key[0] = current.key[current.size]; newRoot.ptr[0] = current; newRoot.ptr[1] = newInternal; newRoot.isLeaf = false; newRoot.size = 1; this.root = newRoot; } else { // Propagate the split upwards this.shiftLevel(current.key[current.size], this.findParent(this.root, current), newInternal); } } } search(x) { // Search for a key in the B-tree if (this.root === null) { return -1; // B-tree is empty } else { let current = this.root; while (current.isLeaf === false) { for (let i = 0; i < current.size; i++) { if (x < current.key[i]) { current = current.ptr[i]; break; } if (i === current.size - 1) { current = current.ptr[i + 1]; break; } } } for (let i = 0; i < current.size; i++) { if (current.key[i] === x) { return 1; // Key found } } return 0; // Key not found } } display(current) { // Display the B-tree if (current === null) { return; } let q = [current]; while (q.length > 0) { let l = q.length; for (let i = 0; i < l; i++) { let tNode = q.shift(); for (let j = 0; j < tNode.size; j++) { if (tNode !== null) { console.log(tNode.key[j] + ' '); } } for (let j = 0; j < tNode.size + 1; j++) { if (tNode.ptr[j] !== null) { q.push(tNode.ptr[j]); } } console.log('\t'); } console.log('\n'); } } findParent(current, child) { // Helper method to find the parent of a given node let parent; if (current.isLeaf || current.ptr[0].isLeaf) { return null; // No parent for leaf nodes } for (let i = 0; i < current.size + 1; i++) { if (current.ptr[i] === child) { parent = current; return parent; } else { parent = this.findParent(current.ptr[i], child); if (parent !== null) { return parent; } } } return parent; } } const bucketSize = 3; // Set the bucket size for the B-tree const btree = new BTree(); // Create a new B-tree console.log('The size of bucket is ' + bucketSize + '!'); // Insert elements into the B-tree btree.insert(1); btree.insert(2); btree.insert(3); btree.display(btree.getRoot()); btree.insert(4); btree.insert(5); btree.display(btree.getRoot()); OutputThe size of bucket is 3! 1 2 3 3 1 2 3 4 5
Time Complexity:
- Insertion: O(log (h*bucketSize)), where h is height of the tree, and bucketSize denotes the number of elements that can be stored in a single bucket.
- Deletion: O(log (h*bucketSize))
Auxiliary Space: O(n), n-> number of elements in the tree.
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