Graph Valid Tree
A medium Graph interview guide. Task statement, worked examples, intuition, and step-by-step solutions.
- Topic
- Graph
- Sheets
- 1
- Core for
- 12 roles
The problem
Given n nodes labeled from 0 to n-1 and a list of undirected edges, determine whether the graph is a valid tree. A valid tree is connected and has no cycles.
Example 1
- Input
- n = 5, edges = [[0,1],[0,2],[0,3],[1,4]]
- Output
- true
- Why
- The graph is connected and has no cycles, so it is a valid tree.
Example 2
- Input
- n = 5, edges = [[0,1],[1,2],[2,3],[1,3],[1,4]]
- Output
- false
- Why
- The graph contains a cycle 1-2-3-1, so it is not a valid tree.
Constraints
- 1 <= n <= 2000
- 0 <= edges.length <= 5000
How to think about it
Updated 2026-09-29A valid tree with n vertices must satisfy two strict structural invariants: it must contain exactly n - 1 edges, and it must be fully connected. If edge count is not n - 1, it is immediately impossible (either disconnected or containing a cycle). Checking connectivity from node 0 after verifying edge count settles the decision in a single traversal.
Approaches, worst first
Edge count plus DSU cycle detection
time O(n * α(n)) · space O(n)
Check `edges.length === n - 1`. If true, run DSU over each edge. If any edge connects two nodes with the same parent, a cycle exists. If all n - 1 unions succeed without a cycle, the graph is a tree.
Edge count plus one traversalWrite this one
time O(n) · space O(n)
First verify `edges.length === n - 1`. Build adjacency lists and run a standard BFS or DFS starting from node 0, keeping a visited set. Return true if and only if the visited set reaches size n.
Where people lose marks · 3
- A graph can have n - 1 edges and still be invalid if it contains both a disconnected isolated node and a cycle elsewhere; verifying visited count reaches n is mandatory.
- When n = 1 and edges = [], the graph is a valid tree with 0 edges; returning false on empty edges breaks this base case.
- Undirected edges added as bidirectional entries in DFS can mistakenly report trivial 2-node cycles unless the caller passes the parent node to ignore reverse traversals.
The theory behind it
Graph — the ground this problem stands on. All Graph problems
What Graph is
A graph is a network of individual points, called vertices or nodes, connected by lines called edges. Think of a subway transit map, an electrical circuit, or a web of social friends. Unlike a tree, a graph has no designated top node and no parent-child hierarchy. Connections can run one-way or both ways, and paths can loop back on themselves to form closed cycles.
When to reach for it
Reach for graph algorithms when inputs describe relationships, networks, flights between cities, course prerequisites, or clone networks. Signals include finding the shortest route across unweighted connections, ordering tasks that depend on earlier tasks, counting isolated clusters, or checking whether a path contains an infinite loop. Whenever problems present pairs of related entities and ask for reachability, distances, or dependencies, graph representations apply.
How the pattern works
First convert edge lists into an adjacency list, mapping each node to an array of its neighbors. Choose your exploration strategy based on the goal: use a queue and breadth-first search to find the shortest path in unweighted networks, or use recursion and depth-first search to explore full paths and detect cycles. Because graphs can have loops, always track visited nodes in a set or boolean array. Add nodes to the visited set at the moment they enter the queue so they are never visited twice.
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Target Roles
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