Binary Search Trees (BST)

Mind Map Summary

• Topic: Binary Search Trees (BST)
• 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.
• Time Complexity:
  • Search: O(log n) on average, O(n) in the worst case (unbalanced tree).
  • Insertion: O(log n) on average, O(n) in the worst case.
  • Deletion: O(log n) on average, O(n) in the worst case.

Practice Exercise

Implement search, insert, and delete operations for a BST. Write a function to validate if a binary tree is a valid BST. Find the lowest common ancestor (LCA) of two nodes in a BST.

Answer

1. BST Operations:

public class TreeNode {
    public int val;
    public TreeNode left;
    public TreeNode right;
    public TreeNode(int val=0, TreeNode left=null, TreeNode right=null) {
        this.val = val;
        this.left = left;
        this.right = right;
    }
}

// Search
public TreeNode SearchBST(TreeNode root, int val) {
    if (root == null || root.val == val) return root;
    return val < root.val ? SearchBST(root.left, val) : SearchBST(root.right, val);
}

// Insert
public TreeNode InsertIntoBST(TreeNode root, int val) {
    if (root == null) return new TreeNode(val);
    if (val < root.val) root.left = InsertIntoBST(root.left, val);
    else root.right = InsertIntoBST(root.right, val);
    return root;
}

// Delete
public TreeNode DeleteNode(TreeNode root, int key) {
    if (root == null) return null;
    if (key < root.val) {
        root.left = DeleteNode(root.left, key);
    } else if (key > root.val) {
        root.right = DeleteNode(root.right, key);
    } else {
        if (root.left == null) return root.right;
        if (root.right == null) return root.left;
        TreeNode minNode = FindMin(root.right);
        root.val = minNode.val;
        root.right = DeleteNode(root.right, root.val);
    }
    return root;
}

private TreeNode FindMin(TreeNode node) {
    while (node.left != null) {
        node = node.left;
    }
    return node;
}

2. Validate BST:

public bool IsValidBST(TreeNode root) {
    return IsValidBST(root, long.MinValue, long.MaxValue);
}

private bool IsValidBST(TreeNode root, long min, long max) {
    if (root == null) return true;
    if (root.val <= min || root.val >= max) return false;
    return IsValidBST(root.left, min, root.val) && IsValidBST(root.right, root.val, max);
}

3. Lowest Common Ancestor (LCA) of a BST:

public TreeNode LowestCommonAncestor(TreeNode root, TreeNode p, TreeNode q) {
    if (p.val < root.val && q.val < root.val) {
        return LowestCommonAncestor(root.left, p, q);
    } else if (p.val > root.val && q.val > root.val) {
        return LowestCommonAncestor(root.right, p, q);
    } else {
        return root;
    }
}