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Graph Valid Tree

GraphsMedium1:30

Given n nodes labeled 0 to n-1 and a list of undirected edges, determine if these edges form a valid tree.

A valid tree has two properties: 1. It is fully connected (all nodes are reachable from any node). 2. It has no cycles.

For example:

n = 5, edges = [[0,1], [0,2], [0,3], [1,4]] -> Valid tree
n = 5, edges = [[0,1], [1,2], [2,3], [1,3], [1,4]] -> Not a valid tree (has a cycle: 1-2-3-1)

Is there a quick check you can do before even traversing the graph?

Explain your approach and analyze the time and space complexity.

How to approach it

  • Hint 1

    A tree with n nodes has exactly n-1 edges. If the edge count is not n-1, it is definitely not a tree.

  • Hint 2

    With exactly n-1 edges, you still need to verify connectivity (no disconnected components).

  • Hint 3

    Use BFS/DFS from any node and check if all n nodes are visited. With n-1 edges and full connectivity, there can be no cycles.

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