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Graph Valid Tree

Medium
Union Find

You have a graph of n nodes labeled from 0 to n - 1. You are given an integer n and a list of edges where edges[i] = [ai, bi] indicates that there is an undirected edge between nodes ai and bi in the graph.

Return true if the edges of the given graph make up a valid tree, and false otherwise.

Tree Properties:

  1. A tree with n nodes has exactly n-1 edges
  2. A tree is connected (all nodes reachable from any node)
  3. A tree has no cycles

Union Find Approach:

  1. Check if number of edges equals n-1 (necessary condition)
  2. Process each edge with Union Find
  3. If find(a) == find(b) before union, there's a cycle - not a tree!
  4. After all edges, check if all nodes are in the same component

Key Insight: A valid tree is a connected graph with no cycles. Union Find detects cycles and connectivity simultaneously.

Example 1

Input: n = 5, edges = [[0,1],[0,2],[0,3],[1,4]]
Output: true
Explanation: The graph is connected with 4 edges (n-1) and no cycles. It is a valid tree.

Example 2

Input: n = 5, edges = [[0,1],[1,2],[2,3],[1,3],[1,4]]
Output: false
Explanation: There is a cycle: 1-2-3-1. Trees cannot have cycles.

Example 3

Input: n = 5, edges = [[0,1],[2,3]]
Output: false
Explanation: The graph is not connected. Node 4 is isolated, and {0,1} and {2,3} are separate components.

Constraints

  • •1 <= n <= 2000
  • •0 <= edges.length <= 5000
  • •edges[i].length == 2
  • •0 <= ai, bi < n
  • •ai != bi
  • •There are no self-loops or repeated edges.
Show Hints (4)
Hint 1: A tree with n nodes must have exactly n-1 edges. Is this sufficient to guarantee a valid tree?
Hint 2: If you have n-1 edges but the graph is disconnected, can it be a tree?
Hint 3: Use Union Find to process edges. What does it mean if find(a) == find(b) before we union them?
Hint 4: After processing all edges, how many connected components should a valid tree have?
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