Graph Valid Tree
Time O(\ · Space V\ · Official statement on LeetCode
Solutions
// Time: O(|V| + |E|)
// Space: O(|V| + |E|)
// Same complexity, but faster version.
class Solution {
public:
bool validTree(int n, vector<pair<int, int>>& edges) {
if (edges.size() != n - 1) {
return false;
}
unordered_map<int, vector<int>> neighbors;
for (const auto& edge : edges) {
neighbors[edge.first].emplace_back(edge.second);
neighbors[edge.second].emplace_back(edge.first);
}
queue<int> q;
q.emplace(0);
unordered_set<int> visited;
visited.emplace(0);
while (!q.empty()) {
const int i = q.front();
q.pop();
for (const auto& node : neighbors[i]) {
if (!visited.count(node)) {
visited.emplace(node);
q.emplace(node);
}
}
}
return visited.size() == n;
}
};
// Time: O(|V| + |E|)
// Space: O(|V| + |E|)
class Solution2 {
public:
struct node {
int parent = -1;
vector<int>neighbors;
};
bool validTree(int n, vector<pair<int, int>>& edges) {
unordered_map<int, node> nodes;
for (const auto& edge : edges) {
nodes[edge.first].neighbors.emplace_back(edge.second);
nodes[edge.second].neighbors.emplace_back(edge.first);
}
queue<int> q;
q.emplace(0);
unordered_set<int> visited;
visited.emplace(0);
while (!q.empty()) {
const int i = q.front();
q.pop();
for (const auto& node : nodes[i].neighbors) {
if (node != nodes[i].parent) {
if (visited.find(node) != visited.end()) {
return false;
} else {
visited.emplace(node);
nodes[node].parent = i;
q.emplace(node);
}
}
}
}
return visited.size() == n;
}
};
Beginner Explanation
What is Graph Valid Tree?
Graph Valid Tree (LeetCode #261) is a Medium problem that primarily trains breadth first search.
How to think about it
- Restate the goal in your own words before coding.
- Work a tiny example by hand so the invariant becomes obvious.
- Identify the pattern — this problem aligns with queue bfs.
- Only then translate the idea into code.
Why this problem matters
It sits in the sweet spot of interview difficulty: multiple valid approaches, clear trade-offs. Official solution notes mention: )_.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Graph Valid Tree
Opening (30–60 seconds)
- Clarify inputs/outputs and edge cases (empty input, single element, duplicates, overflow).
- State a brute force so the interviewer knows you can solve it naively.
- Propose the optimal direction tied to queue bfs.
Core solution narrative
- Define the state you track (pointers, DP cell, set membership, stack top, etc.).
- Explain the transition when you process the next element.
- Call out time (O() and space (V) before coding.
- Code cleanly; narrate variable names.
What interviewers listen for
- Correctness on edge cases
- Complexity honesty
- Ability to discuss trade-offs (e.g., hash map space vs. sort + two pointers)
Follow-up questions they may ask
- Can you solve it with less memory?
- What if the input stream is infinite / doesn't fit in RAM?
- How would tests look for adversarial inputs?
Optimized Approach
Optimized solution notes
The reference solutions on AlgoForge target *O(* time and *V* space.
Pattern focus: queue bfs
Use the pattern as a checklist:
- queue bfs — confirm the invariant holds after each step
Multiple methods appear in the source solutions — compare them and explain when each is preferable.
Implementation tips
- Prefer readable names over micro-optimizations in interviews.
- Extract helpers only when they clarify (e.g., expand-around-center, DFS visit).
- After AC-level logic, re-scan for off-by-one and null checks.
Complexity Analysis
Complexity
| Measure | Bound |
|---|---|
| Time | *O(* |
| Space | *V* |
How to justify this in an interview
- Time: count loops, map/set operations, and recursive branching; state average vs worst case if relevant.
- Space: include hash maps, recursion stack, and output allocation when the problem asks for it.
If your implementation differs from the reference, re-derive big-O from your code — never memorize a complexity you cannot defend.
Common Mistakes
Common mistakes on Graph Valid Tree
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for queue bfs — updating state too early or too late.
- Mutating input unexpectedly when the problem forbids it.
- Off-by-one in windows, ranges, or binary search bounds.
- Ignoring overflow / precision for integer arithmetic problems.
- Overengineering — jumping to an advanced structure when a simpler approach works.
Alternative Approaches
Alternatives
The source file includes more than one method. Compare:
- Primary optimized path — best complexity for typical interviews.
- Secondary approach — often brute force, sorting-based, or space-optimized variant.
Practice articulating when you would pick each (constraints, readability, follow-ups).
Edge Cases
Edge cases checklist
- Minimum input size
- Maximum input size / time limits
- Duplicates and already-sorted input
- Negative numbers / zeros (if applicable)
- Disconnected structures (graphs/trees)
- Single path vs branching recursion depth
Pattern Recognition
Spotting this pattern
Signal phrases that point to queue bfs:
- Sorted input or ability to sort without changing the answer class
- Need for contiguous subarray / substring → consider sliding window
- Need for O(1) membership → hash set/map
- Optimal substructure + overlapping subproblems → DP
- Connectivity / components → graph DFS/BFS or Union-Find
Primary topics: breadth first search.
Follow-up Interview Questions
Follow-ups
- How does the solution change if the input is a stream?
- Can you solve it in-place?
- What if duplicates must be handled differently?
- How would you parallelize the approach?
- Design tests that would break a buggy implementation.
Practice Recommendations
What to practice next
- Re-solve Graph Valid Tree in a second language (cpp, python).
- Drill 3–5 more problems tagged breadth first search.
- Teach the solution out loud in under 5 minutes.
- Add this problem to your revision calendar in 3 days and 14 days.
Visualization
Study checklist
- Read the official problem statement on LeetCode
- Solve on paper / whiteboard first
- Implement the queue bfs approach
- Verify edge cases from the checklist
- State time and space complexity aloud
- Compare with the AlgoForge reference solution
- Schedule a revision session
Revision notes
Graph Valid Tree (#261) — Medium. Pattern: queue bfs. Complexity: O(\ time / V\ space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Graph Valid Tree?+
The reference solutions aim for O(\ time and V\ space. Always re-derive complexity from the code you write in the interview.
What pattern does Graph Valid Tree use?+
It primarily maps to queue bfs, within the broader topic of breadth first search.
Is Graph Valid Tree good for interviews?+
Yes — as a Medium problem it is a solid practice target. Pair it with related problems in the same pattern family for spaced repetition.
Where can I read the official statement?+
Open the official LeetCode page for constraints and examples: https://leetcode.com/problems/graph-valid-tree/