Power Tree

You are given a rooted tree with $$$n$$$ vertices, the root of the tree is the vertex $$$1$$$. Each vertex has some non-negative price. A leaf of the tree is a non-root vertex that has degree $$$1$$$.

Arkady and Vasily play a strange game on the tree. The game consists of three stages. On the first stage Arkady buys some non-empty set of vertices of the tree. On the second stage Vasily puts some integers into all leaves of the tree. On the third stage Arkady can perform several (possibly none) operations of the following kind: choose some vertex $$$v$$$ he bought on the first stage and some integer $$$x$$$, and then add $$$x$$$ to all integers in the leaves in the subtree of $$$v$$$. The integer $$$x$$$ can be positive, negative of zero.

A leaf $$$a$$$ is in the subtree of a vertex $$$b$$$ if and only if the simple path between $$$a$$$ and the root goes through $$$b$$$.

Arkady's task is to make all integers in the leaves equal to zero. What is the minimum total cost $$$s$$$ he has to pay on the first stage to guarantee his own win independently of the integers Vasily puts on the second stage? Also, we ask you to find all such vertices that there is an optimal (i.e. with cost $$$s$$$) set of vertices containing this one such that Arkady can guarantee his own win buying this set on the first stage.