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Binary Search Tree 

The following is the definition of Binary Search Tree (BST) according to Wikipedia
Binary Search Tree is a node-based binary tree data structure which has the following properties:  

• The left subtree of a node contains only nodes with keys lesser than the node’s key.
• The right subtree of a node contains only nodes with keys greater than the node’s key.
• The left and right subtree each must also be a binary search tree. 
  There must be no duplicate
  

  The above properties of Binary Search Tree provides an ordering among keys so that the operations like search, minimum and maximum can be done fast. If there is no ordering, then we may have to compare every key to search for a given key.

  Searching a key 
  For searching a value, if we had a sorted array we could have performed a binary search. Let’s say we want to search a number in the array, in binary search, we first define the complete list as our search space, the number can exist only within the search space. Now we compare the number to be searched or the element to be searched with the middle element (median) of the search space and if the record being searched is less than the middle element, we go searching in the left half, else we go searching in the right half, in case of equality we have found the element. In binary search we start with ‘n’ elements in search space and if the mid element is not the element that we are looking for, we reduce the search space to ‘n/2’ we keep reducing the search space until we either find the record that we are looking for or we get to only one element in search space and be done with this whole reduction. 

  Search operations in binary search tree will be very similar. Let’s say we want to search for the number, we start at the root, and then we compare the value to be searched with the value of the root, if it’s equal we are done with the search if it’s smaller we know that we need to go to the left subtree because in a binary search tree all the elements in the left subtree are smaller and all the elements in the right subtree are larger. Searching an element in the binary search tree is basically this traversal, at each step we go either left or right and at each step we discard one of the sub-trees. If the tree is balanced (we call a tree balanced if for all nodes the difference between the heights of left and right subtrees is not greater than one) we start with a search space of ‘n’ nodes and as we discard one of the sub-trees, we discard ‘n/2’ nodes so our search space gets reduced to ‘n/2’. In the next step we reduce the search space to ‘n/4’ and we repeat until we find the element or our search space is reduced to only one node. The search here is also a binary search hence the name; Binary Search Tree.

  Try It

  • C++
  • Java
  • Python
  • C#
  • Javascript

  // C function to search a given key in a given BST struct node* search(struct node* root, int key) {     // Base Cases: root is null or key is present at root     if (root == NULL || root->key == key)        return root;          // Key is greater than root's key     if (root->key < key)        return search(root->right, key);       // Key is smaller than root's key     return search(root->left, key); }

  Illustration to search 6 in below tree: 
  1. Start from the root. 
  2. Compare the searching element with root, if less than root, then recursively call left subtree, else recursively call right subtree. 
  3. If the element to search is found anywhere, return true, else return false. 
   

  

   
  Insertion of a key 
  A new key is always inserted at the leaf. We start searching a key from the root until we hit a leaf node. Once a leaf node is found, the new node is added as a child of the leaf node. 
   

           100                               100
          /   \        Insert 40            /    \
        20     500    --------->          20     500 
       /  \                              /  \  
      10   30                           10   30
                                                \   
                                                40

  • C++
  • C
  • Java
  • Python
  • C#
  • Javascript

  // C++ program to demonstrate insertion // in a BST recursively. #include <iostream> using namespace std;   class BST {     int data;     BST *left, *right;   public:     // Default constructor.     BST();       // Parameterized constructor.     BST(int);       // Insert function.     BST* Insert(BST*, int);       // Inorder traversal.     void Inorder(BST*); };   // Default Constructor definition. BST ::BST()     : data(0)     , left(NULL)     , right(NULL) { }   // Parameterized Constructor definition. BST ::BST(int value) {     data = value;     left = right = NULL; }   // Insert function definition. BST* BST ::Insert(BST* root, int value) {     if (!root) {         // Insert the first node, if root is NULL.         return new BST(value);     }       // Insert data.     if (value > root->data) {         // Insert right node data, if the 'value'         // to be inserted is greater than 'root' node data.           // Process right nodes.         root->right = Insert(root->right, value);     }     else {         // Insert left node data, if the 'value'         // to be inserted is greater than 'root' node data.           // Process left nodes.         root->left = Insert(root->left, value);     }       // Return 'root' node, after insertion.     return root; }   // Inorder traversal function. // This gives data in sorted order. void BST ::Inorder(BST* root) {     if (!root) {         return;     }     Inorder(root->left);     cout << root->data << endl;     Inorder(root->right); }   // Driver code int main() {     BST b, *root = NULL;     root = b.Insert(root, 50);     b.Insert(root, 30);     b.Insert(root, 20);     b.Insert(root, 40);     b.Insert(root, 70);     b.Insert(root, 60);     b.Insert(root, 80);       b.Inorder(root);     return 0; }   // This code is contributed by pkthapa

  Output

  20
  30
  40
  50
  60
  70
  80

  Illustration to insert 2 in below tree: 
  1. Start from the root. 
  2. Compare the inserting element with root, if less than root, then recursively call left subtree, else recursively call right subtree. 
  3. After reaching the end, just insert that node at left(if less than current) else right. 
   

  

  Time Complexity: The worst-case time complexity of search and insert operations is O(h) where h is the height of the 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 O(n). 

  Insertion using loop:

  • Java
  • C#

  import java.util.*; import java.io.*;   class GFG {     public static void main (String[] args) {          BST tree=new BST();         tree.insert(30);         tree.insert(50);         tree.insert(15);         tree.insert(20);         tree.insert(10);         tree.insert(40);         tree.insert(60);         tree.inorder();     } }   class Node{     Node left;     int val;     Node right;     Node(int val){         this.val=val;     } }   class BST{   Node root;       public void insert(int key){         Node node=new Node(key);         if(root==null) {             root = node;             return;         }         Node prev=null;         Node temp=root;         while (temp!=null){             if(temp.val>key){                 prev=temp;                 temp=temp.left;             }             else if (temp.val<key){                 prev=temp;                 temp=temp.right;             }         }         if(prev.val>key)             prev.left=node;         else prev.right=node;     }        public void inorder(){         Node temp=root;         Stack<Node> stack=new Stack<>();         while (temp!=null||!stack.isEmpty()){             if(temp!=null){                 stack.add(temp);                 temp=temp.left;             }             else {                 temp=stack.pop();                 System.out.print(temp.val+" ");                 temp=temp.right;             }         }     } }

  Output

  10 15 20 30 40 50 60 

   

  Some Interesting Facts:  

  • Inorder traversal of BST always produces sorted output.
  • We can construct a BST with only Preorder or Postorder or Level Order traversal. Note that we can always get inorder traversal by sorting the only given traversal.
  • Number of unique BSTs with n distinct keys is Catalan Number

  Related Links: 

  • Binary Search Tree Delete Operation
  • Quiz on Binary Search Tree
  • Coding practice on BST
  • All Articles on BST

  Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

  
  

  
  

  
  

  
  
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