Let <code>dp[i]</code> be the maximum score we can get on the subtree rooted at <code>i</code> and <code>sum[i]</code> be the sum of all the values of the subtree rooted at <code>i</code>.

If we don’t take <code>value[i]</code> into the final score, we can take all the nodes of the subtrees rooted at <code>i</code>’s children.

If we take <code>value[i]</code> into the score, then each subtree rooted at its children should satisfy the constraints.

<code>dp[x] = max(value[x] + sigma(dp[y]), sigma(sum[y]))</code>, where <code>y</code> is a direct child of <code>x</code>.