Two Sum in BST in DSA C++ - Time & Space Complexity

Time Complexity: Two Sum in BST

O(n)

Understanding Time Complexity

We want to know how the time needed to find two numbers that add up to a target grows as the tree gets bigger.

How does the search time change when the number of nodes in the BST increases?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

bool findTarget(TreeNode* root, int k) {
    unordered_set<int> seen;
    return dfs(root, k, seen);
}

bool dfs(TreeNode* node, int k, unordered_set<int>& seen) {
    if (!node) return false;
    if (seen.count(k - node->val)) return true;
    seen.insert(node->val);
    return dfs(node->left, k, seen) || dfs(node->right, k, seen);
}
    

This code searches the BST to find if any two nodes sum to the target value k using a depth-first search and a set to track visited values.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Visiting each node once in the tree using recursion.
• How many times: Each node is visited exactly one time.

How Execution Grows With Input

As the number of nodes grows, the function checks each node once, so the work grows directly with the number of nodes.

Input Size (n)	Approx. Operations
10	About 10 node visits
100	About 100 node visits
1000	About 1000 node visits

Pattern observation: The operations increase in a straight line with the number of nodes.

Final Time Complexity

Time Complexity: O(n)

This means the time to find the two sum grows directly with the number of nodes in the BST.

Common Mistake

[X] Wrong: "Because the tree is sorted, we can find the two sum in less than linear time like O(log n)."

[OK] Correct: The code visits every node to check pairs, so it still needs to look at all nodes in the worst case.

Interview Connect

Understanding how to analyze recursive tree searches helps you explain your approach clearly and shows you know how the algorithm scales with input size.

Self-Check

"What if we used an in-order traversal to create a sorted list first, then used two pointers to find the sum? How would the time complexity change?"

Practice

1. What is the main goal of the Two Sum in BST problem?

easy

A. Find the maximum value in a BST

B. Find two nodes in a BST whose values add up to a target sum

C. Count the number of nodes in a BST

D. Check if a BST is balanced

Solution

1. Step 1: Understand the problem statement

   The problem asks to find if there exist two nodes in a Binary Search Tree (BST) such that their values add up to a given target sum.

2. Step 2: Identify the goal

   The goal is to check for the existence of such a pair, not to find maximum, count nodes, or check balance.

3. Final Answer:

   Find two nodes in a BST whose values add up to a target sum -> Option B

4. Quick Check:

   Two Sum in BST means finding two nodes summing to target [OK]

Hint: Look for two nodes adding to target in BST [OK]

Common Mistakes:

• Confusing with finding max/min value
• Thinking it counts nodes instead of pairs
• Assuming it checks tree balance

2. Which of the following C++ code snippets correctly declares a BST node structure for the Two Sum problem?

easy

A. struct TreeNode { int val; TreeNode left; TreeNode* right; };

B. struct TreeNode { int val; TreeNode left; TreeNode right; };

C. struct TreeNode { int val; TreeNode* left; TreeNode* right; };

D. struct TreeNode { int val; TreeNode* left; TreeNode right; };

Solution

1. Step 1: Check pointer usage in struct

   In C++, child nodes in a tree are pointers to TreeNode objects, so left and right must be pointers.

2. Step 2: Validate each option

   struct TreeNode { int val; TreeNode* left; TreeNode* right; }; uses TreeNode* for left and right, which is correct. Options A, B, and C incorrectly use TreeNode without pointers for one or both children.

3. Final Answer:

   struct TreeNode { int val; TreeNode* left; TreeNode* right; }; -> Option C

4. Quick Check:

   BST nodes use pointers for children [OK]

Hint: Children in BST nodes are pointers in C++ [OK]

Common Mistakes:

• Using objects instead of pointers for children
• Mixing pointer and non-pointer types
• Forgetting semicolon after struct

3. What is the output of the following code snippet that checks for two sum in BST?

struct TreeNode {
  int val;
  TreeNode* left;
  TreeNode* right;
  TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
};

bool findTarget(TreeNode* root, int k) {
  std::unordered_set<int> seen;
  std::function dfs = [&](TreeNode* node) {
    if (!node) return false;
    if (seen.count(k - node->val)) return true;
    seen.insert(node->val);
    return dfs(node->left) || dfs(node->right);
  };
  return dfs(root);
}

int main() {
  TreeNode* root = new TreeNode(5);
  root->left = new TreeNode(3);
  root->right = new TreeNode(6);
  root->left->left = new TreeNode(2);
  root->left->right = new TreeNode(4);
  root->right->right = new TreeNode(7);
  
  bool result = findTarget(root, 9);
  std::cout << (result ? "true" : "false") << std::endl;
  return 0;
}

medium

A. true

B. false

C. Compilation error

D. Runtime error

Solution

1. Step 1: Understand the tree and target

   The BST has nodes 5,3,6,2,4,7. The target sum is 9.

2. Step 2: Trace the DFS and set usage

   DFS visits nodes and checks if (target - current node value) is in the set. For example, when visiting node 6, it checks if 9-6=3 is in set. Node 3 was visited before, so it returns true.

3. Final Answer:

   true -> Option A

4. Quick Check:

   3 + 6 = 9 found in BST [OK]

Hint: Check if complement exists in visited nodes set [OK]

Common Mistakes:

• Assuming no pair exists without checking all nodes
• Confusing DFS order with BFS
• Ignoring the use of hash set for complements

4. Identify the bug in the following Two Sum in BST code snippet and select the correct fix:

bool findTarget(TreeNode* root, int k) {
  std::unordered_set<int> seen;
  std::function dfs = [&](TreeNode* node) {
    if (!node) return false;
    if (seen.count(k - node->val)) return true;
    dfs(node->left);
    dfs(node->right);
    seen.insert(node->val);
    return false;
  };
  return dfs(root);
}

medium

A. Insert node->val before recursive calls to children

B. Remove the unordered_set and use a vector instead

C. Change return false to return true at the end

D. Add a null check after inserting node->val

Solution

1. Step 1: Analyze the order of operations

   The code inserts node->val into the set after recursive calls, so complements are not found when visiting children.

2. Step 2: Fix the insertion timing

   Insert node->val before recursive calls so that children can find complements in the set.

3. Final Answer:

   Insert node->val before recursive calls to children -> Option A

4. Quick Check:

   Insert before recursion to find complements [OK]

Hint: Insert current value before recursion to detect complements [OK]

Common Mistakes:

• Inserting after recursion causes missed pairs
• Changing return values incorrectly
• Replacing set with vector reduces efficiency

5. Given a BST with nodes [10, 5, 15, 3, 7, null, 18], how would you modify the Two Sum in BST algorithm to handle the case where the target sum is twice the value of a single node (e.g., target = 14 and node value = 7)?

hard

A. Only check leaf nodes for pairs

B. Ignore pairs where complement equals current node value

C. Use a sorted array instead of a hash set

D. Check if the complement equals the current node value and ensure the node appears twice

Solution

1. Step 1: Understand the problem with duplicates

   If target is twice a node's value, we must ensure two distinct nodes with that value exist.

2. Step 2: Modify algorithm to handle duplicates

   When complement equals current node value, check if the node appears twice (e.g., by counting occurrences or checking different nodes).

3. Final Answer:

   Check if the complement equals the current node value and ensure the node appears twice -> Option D

4. Quick Check:

   Handle duplicates by verifying two distinct nodes [OK]

Hint: Verify two distinct nodes if complement equals current value [OK]

Common Mistakes:

• Ignoring duplicate values causing false positives
• Skipping pairs with same node value
• Switching data structures unnecessarily