Andrey's Tree
Contest: Codeforces Round 932 (Div. 2)
Difficulty rating: 2800
Tags: binary search, constructive algorithms, data structures, dfs and similar, dsu, greedy, implementation, trees

Master Andrey loves trees$$$^{\dagger}$$$ very much, so he has a tree consisting of $$$n$$$ vertices.

But it's not that simple. Master Timofey decided to steal one vertex from the tree. If Timofey stole vertex $$$v$$$ from the tree, then vertex $$$v$$$ and all edges with one end at vertex $$$v$$$ are removed from the tree, while the numbers of other vertices remain unchanged. To prevent Andrey from getting upset, Timofey decided to make the resulting graph a tree again. To do this, he can add edges between any vertices $$$a$$$ and $$$b$$$, but when adding such an edge, he must pay $$$|a - b|$$$ coins to the Master's Assistance Center.

Note that the resulting tree does not contain vertex $$$v$$$.

Timofey has not yet decided which vertex $$$v$$$ he will remove from the tree, so he wants to know for each vertex $$$1 \leq v \leq n$$$, the minimum number of coins needed to be spent to make the graph a tree again after removing vertex $$$v$$$, as well as which edges need to be added.

$$$^{\dagger}$$$A tree is an undirected connected graph without cycles.

Input format:
Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$5 \le n \le 2\cdot10^5$$$) — the number of vertices in Andrey's tree.

The next $$$n - 1$$$ lines contain a description of the tree's edges. The $$$i$$$-th of these lines contains two integers $$$u_i$$$ and $$$v_i$$$ ($$$1 \le u_i, v_i \le n$$$) — the numbers of vertices connected by the $$$i$$$-th edge.

It is guaranteed that the given edges form a tree.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2\cdot10^5$$$.

Output format:
For each test case, output the answer in the following format:

For each vertex $$$v$$$ (in the order from $$$1$$$ to $$$n$$$), in the first line output two integers $$$w$$$ and $$$m$$$ — the minimum number of coins that need to be spent to make the graph a tree again after removing vertex $$$v$$$, and the number of added edges.

Then output $$$m$$$ lines, each containing two integers $$$a$$$ and $$$b$$$ ($$$1 \le a, b \le n, a \ne v, b \ne v$$$, $$$a \ne b$$$) — the ends of the added edge.

If there are multiple ways to add edges, you can output any solution with the minimum cost.

Constraints:
Time limit: 4.0 s
Memory limit: 256.0 MB

Note:
In the first test case:

Consider the removal of vertex $$$4$$$:

The optimal solution would be to add an edge from vertex $$$5$$$ to vertex $$$3$$$. Then we will spend $$$|5 - 3| = 2$$$ coins.

In the third test case:

Consider the removal of vertex $$$1$$$:

The optimal solution would be:

- Add an edge from vertex $$$2$$$ to vertex $$$3$$$, spending $$$|2 - 3| = 1$$$ coin.
- Add an edge from vertex $$$3$$$ to vertex $$$4$$$, spending $$$|3 - 4| = 1$$$ coin.
- Add an edge from vertex $$$4$$$ to vertex $$$5$$$, spending $$$|4 - 5| = 1$$$ coin.

Then we will spend a total of $$$1 + 1 + 1 = 3$$$ coins.

Consider the removal of vertex $$$2$$$:

No edges need to be added, as the graph will remain a tree after removing the vertex.

Example input:
3
5
1 3
1 4
4 5
3 2
5
4 2
4 3
3 5
5 1
5
2 1
1 5
1 4
1 3

Example output:
1 1
3 4

0 0

1 1
1 2

2 1
3 5

0 0

0 0

0 0

1 1
1 2

1 1
1 2

1 1
1 2

3 3
2 3
4 5
3 4

0 0

0 0

0 0

0 0

Editorial: