A14893. Equal Tree Sums

You are given an undirected unrooted tree, i.e. a connected undirected graph without cycles.

You must assign a nonzero integer weight to each vertex so that the following is satisfied: if any vertex of the tree is removed, then each of the remaining connected components has the same sum of weights in its vertices.

The input consists of multiple test cases. The first line contains a single integer $t$ ( $1 \leq t \leq 10^4$ ) — the number of test cases. Description of the test cases follows.

The first line of each test case contains an integer $n$ ( $3 \leq n \leq 10^5$ ) — the number of vertices of the tree.

The next $n-1$ lines of each case contain each two integers $u, v$ ( $1 \leq u,v \leq n$ ) denoting that there is an edge between vertices $u$ and $v$ . It is guaranteed that the given edges form a tree.

The sum of $n$ for all test cases is at most $10^5$ .

For each test case, you must output one line with $n$ space separated integers $a_1, a_2, \ldots, a_n$ , where $a_i$ is the weight assigned to vertex $i$ . The weights must satisfy $-10^5 \leq a_i \leq 10^5$ and $a_i \neq 0$ .

It can be shown that there always exists a solution satisfying these constraints. If there are multiple possible solutions, output any of them.

In the first case, when removing vertex $1$ all remaining connected components have sum $5$ and when removing vertex $3$ all remaining connected components have sum $2$ . When removing other vertices, there is only one remaining connected component so all remaining connected components have the same sum.