261 · Graph Valid Tree

Given n nodes labeled 0 to n - 1 and a list of undirected edges. Determine whether these edges make up a valid tree.

Tree Axiom Evaluation

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INVARIANT Tree Equivalence Invariant: For any undirected graph with n nodes, the following is equivalent: (1) Graph is connected and acyclic ⇔ (2) edges.length == n - 1 AND graph is fully connected. If edges.length != n - 1, immediately reject in O(1).

Axiom 1 (edges == n - 1): -

Axiom 2 (all connected): -

Edges:0

Visited Nodes:0 / 0

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Java 21 Solution

INTERVIEW SCRIPT

How to Explain Graph Valid Tree to an Interviewer

1. The 2-Axiom Simplification

"A graph is a tree if and only if it is connected and has no cycles. Graph theory proves: for $n$ nodes, having $n-1$ edges and being fully connected automatically implies 0 cycles!"

2. O(1) Edge Gate Check

"If edges.length != n - 1, return false immediately! Fewer edges means disconnected components; more edges means guaranteed cycles."

3. Single-Component Reachability

"Since edges.length == n - 1 is verified, we only need to check if all $n$ nodes belong to 1 connected component. Run BFS/DFS from node 0 and check visitedCount == n."

4. Complexity Analysis

"Because we gate on $E = n - 1$, BFS visits at most $n$ vertices and $n-1$ edges. Total time is strictly $O(n)$, and auxiliary space is $O(n)$."