B-Tree Insert without aggressive splitting

This algorithm for insertion takes an entry, finds the leaf node where it belongs, and inserts it there. We recursively insert the entry by calling the insert algorithm on the appropriate child node. This procedure results in going down to the leaf node where the entry belongs, placing the entry there, and returning all the way back to the root node.

Sometimes a node is full, i.e. it contains 2*t – 1 entries where t is the minimum degree. In such cases the node must be split. In such case one key becomes a parent and a new node is created. We first insert the new key, making total keys as 2*t. We keep the first t entries in original node, transfer last (t-1) entries to new node and set the (t+1)th node as parent of these nodes. If the node being split is a non child node then we also have to split the child pointers. A node having 2*t keys has 2*t + 1 child pointers. The first (t+1) pointers are kept in original node and remaining t pointers goes to new node.

This algorithm splits a node only when it is necessary. We first recursively call insert for appropriate child of node (in case of non-leaf node) or insert it into node (for leaf node). If the node is full, we split it, storing new child entry in newEntry and new parent key in val. These values are then inserted into the parent, which recursively splits itself in case it is also full.

Example:

We insert numbers 1 – 5 in tree. The tree becomes:

Then we insert 6, the node is full. Hence it is split into two nodes making 4 as parent.

We insert numbers 7 – 16, the tree becomes:

We insert 22 – 30, the tree becomes:

Note that now the root is full. If we insert 17 now, the root is not split as the leaf node in which 17 was inserted didn’t split. If we were following aggressive splitting, the root would have been split before we went to leaf node.

But if we insert 31, the leaf node splits, which recursively adds new entry to root. But as root is full, it needs to be split. The tree now becomes.

Below is the implementation of the above approach:

// C++ implementation of the approach 
#include <iostream> 
#include <vector> 
using namespace std; 
  
class BTreeNode { 
  
    // Vector of keys 
    vector<int> keys; 
  
    // Minimum degree 
    int t; 
  
    // Vector of child pointers 
    vector<BTreeNode*> C; 
  
    // Is true when node is leaf, else false 
    bool leaf; 
  
public: 
    // Constructor 
    BTreeNode(int t, bool leaf); 
  
    // Traversing the node and print its content 
    // with tab number of tabs before 
    void traverse(int tab); 
  
    // Insert key into given node. If child is split, we 
    // have to insert *val entry into keys vector and 
    // newEntry pointer into C vector of this node 
    void insert(int key, int* val, 
                BTreeNode*& newEntry); 
  
    // Split this node and store the new parent value in 
    // *val and new node pointer in newEntry 
    void split(int* val, BTreeNode*& newEntry); 
  
    // Returns true if node is full 
    bool isFull(); 
  
    // Makes new root, setting current root as its child 
    BTreeNode* makeNewRoot(int val, BTreeNode* newEntry); 
}; 
  
bool BTreeNode::isFull() 
{ 
    // returns true if node is full 
    return (this->keys.size() == 2 * t - 1); 
} 
  
BTreeNode::BTreeNode(int t, bool leaf) 
{ 
    // Constructor to set value of t and leaf 
    this->t = t; 
    this->leaf = leaf; 
} 
  
// Function to print the nodes of B-Tree 
void BTreeNode::traverse(int tab) 
{ 
    int i; 
    string s; 
  
    // Print 'tab' number of tabs 
    for (int j = 0; j < tab; j++) { 
        s += '\t'; 
    } 
    for (i = 0; i < keys.size(); i++) { 
  
        // If this is not leaf, then before printing key[i] 
        // traverse the subtree rooted with child C[i] 
        if (leaf == false) 
            C[i]->traverse(tab + 1); 
        cout << s << keys[i] << endl; 
    } 
  
    // Print the subtree rooted with last child 
    if (leaf == false) { 
        C[i]->traverse(tab + 1); 
    } 
} 
  
// Function to split the current node and store the new 
// parent value is *val and new child pointer in &newEntry 
// called only for splitting non-leaf node 
void BTreeNode::split(int* val, BTreeNode*& newEntry) 
{ 
  
    // Create new non leaf node 
    newEntry = new BTreeNode(t, false); 
  
    //(t+1)th becomes parent 
    *val = this->keys[t]; 
  
    // Last (t-1) entries will go to new node 
    for (int i = t + 1; i < 2 * t; i++) { 
        newEntry->keys.push_back(this->keys[i]); 
    } 
  
    // This node stores first t entries 
    this->keys.resize(t); 
  
    // Last t entries will go to new node 
    for (int i = t + 1; i <= 2 * t; i++) { 
        newEntry->C.push_back(this->C[i]); 
    } 
  
    // This node stores first (t+1) entries 
    this->C.resize(t + 1); 
} 
  
// Function to insert a new key in given node. 
// If child of this node is split, we have to insert *val 
// into keys vector and newEntry pointer into C vector 
void BTreeNode::insert(int new_key, int* val, 
                       BTreeNode*& newEntry) 
{ 
  
    // Non leaf node 
    if (leaf == false) { 
        int i = 0; 
  
        // Find first key greater than new_key 
        while (i < keys.size() && new_key > keys[i]) 
            i++; 
  
        // We have to insert new_key into left child of 
        // Node with index i 
        C[i]->insert(new_key, val, newEntry); 
  
        // No split was done 
        if (newEntry == NULL) 
            return; 
        if (keys.size() < 2 * t - 1) { 
  
            // This node can accomodate a new key 
            // and child pointer entry 
            // Insert *val into key vector 
            keys.insert(keys.begin() + i, *val); 
  
            // Insert newEntry into C vector 
            C.insert(C.begin() + i + 1, newEntry); 
  
            // As this node was not split, set newEntry 
            // to NULL 
            newEntry = NULL; 
        } 
        else { 
  
            // Insert *val and newentry 
            keys.insert(keys.begin() + i, *val); 
            C.insert(C.begin() + i + 1, newEntry); 
  
            // Current node has 2*t keys, so split it 
            split(val, newEntry); 
        } 
    } 
    else { 
  
        // Insert new_key in this node 
        vector<int>::iterator it; 
  
        // Find correct position 
        it = lower_bound(keys.begin(), keys.end(), 
                         new_key); 
  
        // Insert in correct position 
        keys.insert(it, new_key); 
  
        // If node is full 
        if (keys.size() == 2 * t) { 
  
            // Create new node 
            newEntry = new BTreeNode(t, true); 
  
            // Set (t+1)th key as parent 
            *val = this->keys[t]; 
  
            // Insert last (t-1) keys into new node 
            for (int i = t + 1; i < 2 * t; i++) { 
                newEntry->keys.push_back(this->keys[i]); 
            } 
  
            // This node stores first t keys 
            this->keys.resize(t); 
        } 
    } 
} 
  
// Function to create a new root 
// setting current node as its child 
BTreeNode* BTreeNode::makeNewRoot(int val, BTreeNode* newEntry) 
{ 
    // Create new root 
    BTreeNode* root = new BTreeNode(t, false); 
  
    // Stores keys value 
    root->keys.push_back(val); 
  
    // Push child pointers 
    root->C.push_back(this); 
    root->C.push_back(newEntry); 
    return root; 
} 
  
class BTree { 
  
    // Root of B-Tree 
    BTreeNode* root; 
  
    // Minimum degree 
    int t; 
  
public: 
    // Constructor 
    BTree(int t); 
  
    // Insert key 
    void insert(int key); 
  
    // Display the tree 
    void display(); 
}; 
  
// Function to create a new BTree with 
// minimum degree t 
BTree::BTree(int t) 
{ 
    root = new BTreeNode(t, true); 
} 
  
// Function to insert a node in the B-Tree 
void BTree::insert(int key) 
{ 
    BTreeNode* newEntry = NULL; 
    int val = 0; 
  
    // Insert in B-Tree 
    root->insert(key, &val, newEntry); 
  
    // If newEntry is not Null then root needs to be 
    // split. Create new root 
    if (newEntry != NULL) { 
        root = root->makeNewRoot(val, newEntry); 
    } 
} 
  
// Prints BTree 
void BTree::display() 
{ 
    root->traverse(0); 
} 
  
// Driver code 
int main() 
{ 
  
    // Create B-Tree 
    BTree* tree = new BTree(3); 
    cout << "After inserting 1 and 2" << endl; 
    tree->insert(1); 
    tree->insert(2); 
    tree->display(); 
  
    cout << "After inserting 5 and 6" << endl; 
    tree->insert(5); 
    tree->insert(6); 
    tree->display(); 
  
    cout << "After inserting 3 and 4" << endl; 
    tree->insert(3); 
    tree->insert(4); 
    tree->display(); 
  
    return 0; 
} 

Output:

After inserting 1 and 2
1
2
After inserting 5 and 6
1
2
5
6
After inserting 3 and 4
    1
    2
    3
4
    5
    6

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