The minimum price sum is always the price of a rooted node.

Let’s root the tree at vertex 0 and find the answer from this perspective.

In the optimal answer maximum price is the sum of the prices of nodes on the path from “u” to “v” where either “u” or “v” is the parent of the second one or neither is a parent of the second one.

The first case is easy to find. For the second case, notice that in the optimal path, “u” and “v” are both leaves. Then we can use dynamic programming to find such a path.

Let DP(v,1) denote “the maximum price sum from node v to leaf, where v is a parent of that leaf” and let DP(v,0) denote “the maximum price sum from node v to leaf, where v is a parent of that leaf - price[leaf]”. Then the answer is maximum of DP(u,0) + DP(v,1) + price[parent] where u, v are directly connected to vertex “parent”.