Why Post-Order?

Post-order DFS visits children before the parent. When you return from recursing on all children, their dp values are ready. You combine them to compute $dp[\text{parent}]$.

function dfs(v, parent)
    for each child in neighbors[v]
        if child ≠ parent then
            dfs(child, v)
    dp[v] := combine children's dp values

The parent check prevents going back up the edge you came from. Without it, you'd treat your parent as a child and recurse forever. Post-order keeps children are processed before their parent. You need child answers to compute the parent answer.