E. 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:\(\{1\},\{2,1\},\{3,1\},\{4,1\},\{3,2,1\},\{4,2,1\},\{4,3,1\},\{4,3,2,1\}\).