Valid Tree Solution

Two conditions to check:

$1.$ Exactly $n - 1$ edges (necessary for any tree)

$2.$ No cycles (every union should succeed)

function validTree(n, edges):
    if edges.length != n - 1:
        return false

    uf = new UnionFind(n)
    for each (u, v) in edges:
        if not uf.union(u, v):
            return false  // cycle detected

    return true

if you have exactly $n-1$ edges and no cycles, the graph must be connected. Why? An acyclic graph with $n$ nodes and $n-1$ edges has exactly one component (any more components would require fewer edges).

Time: $O(n \cdot \alpha(n)) \approx O(n)$. Space: $O(n)$.