Maximize Walk Value

There's a tree and every one of its nodes has a cost associated with it. Some of these nodes are labelled special nodes. You are supposed to answer a few queries on this tree. In each query, a source and destination node ($SNODE$ and $DNODE$) is given along with a value $W$. For a walk between $SNODE$ and $DNODE$ to be valid you have to choose a special node and call it the pivot $P$. Now the path