CF1830D Mex Tree

You are given a tree with $ n $ nodes. For each node, you either color it in $ 0 $ or $ 1 $ . The value of a path $ (u,v) $ is equal to the MEX $ ^\dagger $ of the colors of the nodes from the shortest path between $ u $ and $ v $ . The value of a coloring is equal to the sum of values of all paths $ (u,v) $ such that $ 1 \leq u \leq v \leq n $ . What is the maximum possible value of any coloring of the tree? $ ^{\dagger} $ The MEX (minimum excluded) of an array is the smallest non-negative integer that does not belong to the array. For instance: - The MEX of $ [2,2,1] $ is $ 0 $ , because $ 0 $ does not belong to the array. - The MEX of $ [3,1,0,1] $ is $ 2 $ , because $ 0 $ and $ 1 $ belong to the array, but $ 2 $ does not. - The MEX of $ [0,3,1,2] $ is $ 4 $ because $ 0 $ , $ 1 $ , $ 2 $ , and $ 3 $ belong to the array, but $ 4 $ does not.

Each test contains multiple test cases. The first line of input contains a single integer $ t $ ( $ 1 \le t \le 10^4 $ ) — the number of test cases. The description of test cases follows. The first line of each test case contains a single integer $ n $ ( $ 1 \le n \le 2 \cdot 10^5 $ ) — the number of nodes in the tree. The following $ n-1 $ lines of each test case contains $ 2 $ integers $ a_i $ and $ b_i $ ( $ 1 \leq a_i, b_i \leq n, a_i \neq b_i $ ) — indicating an edge between vertices $ a_i $ and $ b_i $ . It is guaranteed that the given edges form a tree. It is guaranteed that the sum of $ n $ across all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case, print the maximum possible value of any coloring of the tree.

In the first sample, we will color vertex $ 2 $ in $ 1 $ and vertices $ 1,3 $ in $ 0 $ . After this, we consider all paths: - $ (1,1) $ with value $ 1 $ - $ (1,2) $ with value $ 2 $ - $ (1,3) $ with value $ 2 $ - $ (2,2) $ with value $ 0 $ - $ (2,3) $ with value $ 2 $ - $ (3,3) $ with value $ 1 $ We notice the sum of values is $ 8 $ which is the maximum possible.