Subtree with minimum sum of nodes' costs
For a node $i$ with children $j_1,\ldots,j_d$, let us denote the best solution for the subtree rooted at $i$ by $\tau(i)$; you're interested in $\tau(r)$, where $r$ is the root. The function $\tau$ satisfies the recurrence
$$
\tau(i) = \min(0,  \sigma(i) + \tau(j_1) + \cdots + \tau(j_d)).
$$
Indeed, there are two cases to consider. Either we take no vertex, in which the answer is 0. Otherwise, we must take $i$ itself (by the constraint), and then we should take the best solutions for the subtrees rooted at all children of $i$.

Like all such recurrences on trees, this one can be evaluated in linear time in many ways.