AT_arc215_f [ARC215F] Scapus

You are given a tree with $ N $ vertices numbered $ 1 $ to $ N $ . The $ i $ -th edge ( $ 1 \le i \le N - 1 $ ) connects vertices $ A_i $ and $ B_i $ . For a path in this tree, define the score of the path as the distance from the path to the farthest vertex. Here, the distance from a path is defined as the minimum distance to a vertex on the path. A path with the minimum score is called a good path. For each $ k = 1, 2, \ldots, N $ , find the number of good paths with exactly $ k $ vertices. Paths are distinguished only if their vertex sets differ.

The input is given from Standard Input in the following format: > $ N $ $ A_1 $ $ B_1 $ $ A_2 $ $ B_2 $ $ \vdots $ $ A_{N - 1} $ $ B_{N - 1} $

Output $ N $ lines. The $ k $ -th line should contain the number of good paths with exactly $ k $ vertices.

### Sample Explanation 1 The minimum score is $ 1 $ . The good paths are the following six paths. - The path with two vertices connecting vertices $ 2 $ and $ 3 $ - The path with three vertices connecting vertices $ 1 $ and $ 3 $ - The path with three vertices connecting vertices $ 2 $ and $ 5 $ - The path with three vertices connecting vertices $ 3 $ and $ 4 $ - The path with four vertices connecting vertices $ 1 $ and $ 5 $ - The path with four vertices connecting vertices $ 4 $ and $ 5 $ ### Constraints - $ 2 \le N \le 2 \times 10^5 $ - $ 1 \le A_i < B_i \le N $ ( $ 1 \le i \le N - 1 $ ) - The given graph is a tree. - All input values are integers.