MEX Tree

You are given a tree with n nodes, numerated from 0 to n-1. For each k between 0 and n, inclusive, you have to count the number of unordered pairs (u,v), u ≠ v, such that the MEX of all the node labels in the shortest path from u to v (including end points) is k.

The MEX of a sequence of integers is the smallest non-negative integer that does not belong to the sequence.

Input

The first line contains a single integer t (1 ≤ t ≤ 10^4) — the number of test cases.

The first line of each test case contains a single integer n (2 ≤ n ≤ 2 ⋅ 10^{5}).

The next n-1 lines of each test case describe the tree that has to be constructed. These lines contain two integers u and v (0 ≤ u,v ≤ n-1) denoting an edge between u and v (u ≠ v).

It is guaranteed that the given edges form a tree.

It is also guaranteed that the sum of n for all test cases does not exceed 2 ⋅ 10^{5}.

Output

For each test case, print n+1 integers: the number of paths in the tree, such that the MEX of all the node labels in that path is k for each k from 0 to n.

Example

Input

2 4 0 1 0 2 2 3 2 1 0

Output

1 2 1 1 1 0 0 1

Note

1. In example case 1, * For k = 0, there is 1 path that is from 2 to 3 as MEX([2, 3]) = 0. * For k = 1, there are 2 paths that is from 0 to 2 as MEX([0, 2]) = 1 and 0 to 3 as MEX([0, 2, 3]) = 1. * For k = 2, there is 1 path that is from 0 to 1 as MEX([0, 1]) = 2. * For k = 3, there is 1 path that is from 1 to 2 as MEX([1, 0, 2]) = 3 * For k = 4, there is 1 path that is from 1 to 3 as MEX([1, 0, 2, 3]) = 4.
2. In example case 2, * For k = 0, there are no such paths. * For k = 1, there are no such paths. * For k = 2, there is 1 path that is from 0 to 1 as MEX([0, 1]) = 2.

inputFormat

Input

The first line contains a single integer t (1 ≤ t ≤ 10^4) — the number of test cases.

The first line of each test case contains a single integer n (2 ≤ n ≤ 2 ⋅ 10^{5}).

The next n-1 lines of each test case describe the tree that has to be constructed. These lines contain two integers u and v (0 ≤ u,v ≤ n-1) denoting an edge between u and v (u ≠ v).

It is guaranteed that the given edges form a tree.

It is also guaranteed that the sum of n for all test cases does not exceed 2 ⋅ 10^{5}.

outputFormat

Output

For each test case, print n+1 integers: the number of paths in the tree, such that the MEX of all the node labels in that path is k for each k from 0 to n.

Example

Input

2 4 0 1 0 2 2 3 2 1 0

Output

1 2 1 1 1 0 0 1

Note

1. In example case 1, * For k = 0, there is 1 path that is from 2 to 3 as MEX([2, 3]) = 0. * For k = 1, there are 2 paths that is from 0 to 2 as MEX([0, 2]) = 1 and 0 to 3 as MEX([0, 2, 3]) = 1. * For k = 2, there is 1 path that is from 0 to 1 as MEX([0, 1]) = 2. * For k = 3, there is 1 path that is from 1 to 2 as MEX([1, 0, 2]) = 3 * For k = 4, there is 1 path that is from 1 to 3 as MEX([1, 0, 2, 3]) = 4.
2. In example case 2, * For k = 0, there are no such paths. * For k = 1, there are no such paths. * For k = 2, there is 1 path that is from 0 to 1 as MEX([0, 1]) = 2.

样例

2
4
0 1
0 2
2 3
2
1 0

1 2 1 1 1
0 0 1

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