A14339. 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 $(u,v)$ , $u \neq v$ , 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.

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

The first line of each test case contains a single integer $n$ ( $2 \le n \le 2 \cdot 10^{5}$ ).

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

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 $2 \cdot 10^{5}$ .

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 $n$ .

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 $MEX([0, 1]) = 2$ .