Tree Removal

You are given a tree$^{\dagger}$ consisting of $N$ vertices where each vertex $u$ is assigned a value $A_u$. You start with a score of $0$ and perform the following operation as many times as you like: - Pick a vertex $u$ whose current degree$^{\ddagger}$ is **odd**; - Add $A_u$ to your score; - Remove vertex $u$ and all its incident edges from the graph. Find a sequence of vertices to remove