CF2089E Black Cat Collapse

The world of the black cat is collapsing. In this world, which can be represented as a rooted tree with root at node $ 1 $ , Liki and Sasami need to uncover the truth about the world. Each day, they can explore a node $ u $ that has not yet collapsed. After this exploration, the black cat causes $ u $ and all nodes in its subtree to collapse. Additionally, at the end of the $ i $ th day, if it exists, the number $ n-i+1 $ node will also collapse. For each $ i $ from $ 1 $ to $ n $ , determine the number of exploration schemes where Liki and Sasami explore exactly $ i $ days (i.e., they perform exactly $ i $ operations), with the last exploration being at node $ 1 $ . The result should be computed modulo $ 998\,244\,353 $ . Note: It is guaranteed that nodes $ 1 $ to $ n $ can form a "DFS" order of the tree, meaning there exists a depth-first search traversal where the $ i $ th visited node is $ i $ .

The first line contains an integer $ t $ ( $ 1 \le t \le 10 $ ) — the number of test cases. The description of the test cases follows. The first line of each test case contains exactly one number $ n $ ( $ 3 \le n \le 80 $ ). Each of the following $ n - 1 $ lines contains two integers $ u_i $ and $ v_i $ , representing two vertices connected by an edge ( $ 1 \le u_i, v_i \le n $ ). It is guaranteed that the given edges form a tree. It is also guaranteed that the vertices can form a "DFS" traversal order. It is guaranteed that the sum of $ n $ for all test cases does not exceed $ 80 $

For each test case, print $ n $ integers, where the $ i $ th integer represents the number of exploration schemes for exactly $ i $ days, modulo $ 998\,244\,353 $ .

For the first test case, the following operation sequences are legal: $ \{1\},\{2,1\},\{3,1\},\{4,1\},\{3,2,1\},\{4,2,1\},\{4,3,1\},\{4,3,2,1\} $ .