Make and Break on Tree

Alice and Bob play a game on a simple undirected tree with $N$ nodes. The game consists of two steps: 1) Alice selects two nodes such that there is **no direct edge** between them, and connects them with an edge. Note that after this step, the graph is no longer a tree. 2) After Alice's step, Bob selects an edge of this graph and deletes it, such that after deletion the graph becomes a tree agai