Most Profitable Path in a Tree

In a rooted tree where each node has a positive or negative gate amount, Bob moves from a given node toward the root while Alice chooses a path from the root to any leaf; if they visit a node simultaneously the amount is split, otherwise only the first visitor gains/loses it — compute the maximum net income Alice can achieve. The core challenge is computing Bob's arrival times along his path and doing a DFS to maximize Alice's time-adjusted path sum.