There is a tree consisting of (n) vertices. Let a caterpillar be denoted by an integer pair ((p, q)) ((1 \leq p, q \leq n), (p \neq q)): its head is at vertex (p), its tail is at vertex (q), and it dominates all the vertices on the simple path from (p) to (q) (including (p) and (q)). The caterpillar sequence of ((p, q)) is defined as the sequence consisting only of the vertices on the simple path, sorted in the ascending order of the distance to (p). Nora and Aron are taking turns moving the caterpillar , with Nora going first. Both players will be using his or her own optimal strategy: They will play to make himself or herself win ; However, if it is impossible, they will play to prevent the other person from winning (thus, the game will end in a tie ). In Nora's turn, she must choose a vertex (u) adjacent to vertex (p), which is not dominated by the caterpillar , and move all the vertices in it by one edge towards vertex (u)(^{\text{∗}}). In Aron's turn, he must choose a vertex (v) adjacent to vertex (q), which is not dominated by the caterpillar , and move all the vertices in it by one edge towards vertex (v). Note that the moves allowed to the two players are different. Whenever (p) is a leaf(^{\text{†}}), Nora wins (^{\text{‡}}). Whenever (q) is a leaf, Aron wins . If either initially both (p) and (q) are leaves, or after (10^{100}) turns the game has not ended, the result is a tie . Please count the number of integer pairs ((p, q)) with (1 \leq p, q \leq n) and (p \neq q) such that, if the caterpillar is initially ((p, q)), Aron wins the game. (^{\text{∗}})In other words: Let the current caterpillar sequence be (c_1, c_2, \ldots, c_k), then after the move, the new caterpillar sequence becomes (d(u, c_1), d(u, c_2), \ldots, d(u, c_k)). Here, (d(x, y)) is the next vertex on the simple path from (y) to (x). $$$^