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Binary Search Tree

A Binary tree is a tree data structure in which each node has at most two children, which are referred to as the left child and the right child.

For a binary tree to be a binary search tree, the values of all the nodes in the left sub-tree of the root node should be smaller than the root node's value. Also the values of all the nodes in the right sub-tree of the root node should be larger than the root node's value.

a simple binary search tree
a simple binary search tree

Insertion Algorithm

  1. Compare values of the root node and the element to be inserted.
  2. If the value of the root node is larger, and if a left child exists, then repeat step 1 with root = current root's left child. Else, insert element as left child of current root.
  3. If the value of the root node is lesser, and if a right child exists, then repeat step 1 with root = current root's right child. Else, insert element as right child of current root.

Deletion Algorithm

  • Deleting a node with no children: simply remove the node from the tree.
  • Deleting a node with one child: remove the node and replace it with its child.
  • Node to be deleted has two children: Find inorder successor of the node. Copy contents of the inorder successor to the node and delete the inorder successor.
  • Note that: inorder successor can be obtained by finding the minimum value in right child of the node.

Sample Code

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// C program to demonstrate delete operation in binary search tree 
#include<stdio.h>
#include<stdlib.h>

struct node
{
    int key;
    struct node *left, *right;
};

// A utility function to create a new BST node
struct node *newNode(int item)
{
    struct node *temp = (struct node *)malloc(sizeof(struct node));
    temp->key = item;
    temp->left = temp->right = NULL;
    return temp;
}

// A utility function to do inorder traversal of BST
void inorder(struct node *root)
{
    if (root != NULL)
    {
        inorder(root->left);
        printf("%d ", root->key);
        inorder(root->right);
    }
}

/* A utility function to insert a new node with given key in BST */
struct node* insert(struct node* node, int key)
{
    /* If the tree is empty, return a new node */
    if (node == NULL) return newNode(key);

    /* Otherwise, recur down the tree */
    if (key < node->key)
        node->left = insert(node->left, key);
    else
        node->right = insert(node->right, key);

    /* return the (unchanged) node pointer */
    return node;
}

/* Given a non-empty binary search tree, return the node with minimum
   key value found in that tree. Note that the entire tree does not
   need to be searched. */
struct node * minValueNode(struct node* node)
{
    struct node* current = node;

    /* loop down to find the leftmost leaf */
    while (current->left != NULL)
        current = current->left;

    return current;
}

/* Given a binary search tree and a key, this function deletes the key
   and returns the new root */
struct node* deleteNode(struct node* root, int key)
{
    // base case
    if (root == NULL) return root;

    // If the key to be deleted is smaller than the root's key,
    // then it lies in left subtree
    if (key < root->key)
        root->left = deleteNode(root->left, key);

    // If the key to be deleted is greater than the root's key,
    // then it lies in right subtree
    else if (key > root->key)
        root->right = deleteNode(root->right, key);

    // if key is same as root's key, then This is the node
    // to be deleted
    else
    {
        // node with only one child or no child
        if (root->left == NULL)
        {
            struct node *temp = root->right;
            free(root);
            return temp;
        }
        else if (root->right == NULL)
        {
            struct node *temp = root->left;
            free(root);
            return temp;
        }

        // node with two children: Get the inorder successor (smallest
        // in the right subtree)
        struct node* temp = minValueNode(root->right);

        // Copy the inorder successor's content to this node
        root->key = temp->key;

        // Delete the inorder successor
        root->right = deleteNode(root->right, temp->key);
    }
    return root;
}

Time Complexity

The worst case time complexity of search, insert, and deletion operations is \(\mathcal{O}(h)\) where h is the height of Binary Search Tree. In the worst case, we may have to travel from root to the deepest leaf node. The height of a skewed tree may become \(N\) and the time complexity of search and insert operation may become \(\mathcal{O}(N)\). So the time complexity of establishing \(N\) node unbalanced tree may become \(\mathcal{O}(N^2)\) (for example the nodes are being inserted in a sorted way). But, with random input the expected time complexity is \(\mathcal{O}(NlogN)\).

However, you can implement other data structures to establish Self-balancing binary search tree (which will be taught later), popular data structures that implementing this type of tree include:

  • 2-3 tree
  • AA tree
  • AVL tree
  • B-tree
  • Red-black tree
  • Scapegoat tree
  • Splay tree
  • Treap
  • Weight-balanced tree