Are duplicate keys allowed in the definition of binary search trees?

Are Duplicate Keys Allowed in the Definition of Binary Search Trees?

Binary Search Trees (BSTs) are a fundamental data structure used to store ordered data. The typical definition of a BST does not allow duplicate keys, as each key must be unique to maintain the integrity of the tree structure. However, handling duplicates can be necessary in some applications, and there are ways to manage them within the BST framework.

Standard Definition of BST

In a standard Binary Search Tree:

• Left Subtree: All nodes in the left subtree have keys less than the key of the root node.
• Right Subtree: All nodes in the right subtree have keys greater than the key of the root node.
• Unique Keys: Each key in the BST is unique.

This definition ensures that every key can be uniquely identified, making search, insertion, and deletion operations straightforward and efficient.

Handling Duplicates in BST

While the standard BST definition does not allow duplicates, there are several ways to modify the BST structure to handle duplicate keys if required:

1. Allow Duplicates on One Side

You can allow duplicate keys by deciding to place them consistently on one side (usually the left or right).

• Left-Side Placement:
  • Insert duplicates into the left subtree.
  • For example, if the current node key is 10 and the new key is also 10, the duplicate key would be inserted into the left subtree.

class TreeNode {
    int key;
    TreeNode left, right;

    public TreeNode(int item) {
        key = item;
        left = right = null;
    }
}

public class BinarySearchTree {
    TreeNode root;

    BinarySearchTree() { root = null; }

    void insert(int key) {
        root = insertRec(root, key);
    }

    TreeNode insertRec(TreeNode root, int key) {
        if (root == null) {
            root = new TreeNode(key);
            return root;
        }

        if (key <= root.key) {
            root.left = insertRec(root.left, key);
        } else {
            root.right = insertRec(root.right, key);
        }

        return root;
    }

    // Other BST methods...
}

• Right-Side Placement:
  • Insert duplicates into the right subtree.
  • For example, if the current node key is 10 and the new key is also 10, the duplicate key would be inserted into the right subtree.

class TreeNode {
    int key;
    TreeNode left, right;

    public TreeNode(int item) {
        key = item;
        left = right = null;
    }
}

public class BinarySearchTree {
    TreeNode root;

    BinarySearchTree() { root = null; }

    void insert(int key) {
        root = insertRec(root, key);
    }

    TreeNode insertRec(TreeNode root, int key) {
        if (root == null) {
            root = new TreeNode(key);
            return root;
        }

        if (key < root.key) {
            root.left = insertRec(root.left, key);
        } else {
            root.right = insertRec(root.right, key);
        }

        return root;
    }

    // Other BST methods...
}

2. Count Duplicates

Another approach is to store a count of the number of times each key appears instead of allowing duplicates as separate nodes.

class TreeNode {
    int key;
    int count;
    TreeNode left, right;

    public TreeNode(int item) {
        key = item;
        count = 1;
        left = right = null;
    }
}

public class BinarySearchTree {
    TreeNode root;

    BinarySearchTree() { root = null; }

    void insert(int key) {
        root = insertRec(root, key);
    }

    TreeNode insertRec(TreeNode root, int key) {
        if (root == null) {
            root = new TreeNode(key);
            return root;
        }

        if (key == root.key) {
            root.count++;
        } else if (key < root.key) {
            root.left = insertRec(root.left, key);
        } else {
            root.right = insertRec(root.right, key);
        }

        return root;
    }

    // Other BST methods...
}

Summary

• Standard BST Definition: Does not allow duplicate keys to maintain unique key identification.
• Handling Duplicates:
  • Place Duplicates on One Side: Insert duplicates consistently into the left or right subtree.
  • Count Duplicates: Store a count of duplicate keys in each node.

Choosing the appropriate method to handle duplicates depends on the specific requirements of your application. For more in-depth knowledge and practical examples on data structures and other programming concepts, consider exploring Grokking the Coding Interview on DesignGurus.io, which provides comprehensive courses on essential coding and interview techniques.