CF1278E.Tests for problem D

We had a really tough time generating tests for problem D. In order to prepare strong tests, we had to solve the following problem.

Given an undirected labeled tree consisting of $n$ vertices, find a set of segments such that:

1. both endpoints of each segment are integers from $1$ to $2n$ , and each integer from $1$ to $2n$ should appear as an endpoint of exactly one segment;
2. all segments are non-degenerate;
3. for each pair $(i, j)$ such that $i \ne j$ , $i \in [1, n]$ and $j \in [1, n]$ , the vertices $i$ and $j$ are connected with an edge if and only if the segments $i$ and $j$ intersect, but neither segment $i$ is fully contained in segment $j$ , nor segment $j$ is fully contained in segment $i$ .

Can you solve this problem too?

The first line contains one integer $n$ ( $1 \le n \le 5 \cdot 10^5$ ) — the number of vertices in the tree.

Then $n - 1$ lines follow, each containing two integers $x_i$ and $y_i$ ( $1 \le x_i, y_i \le n$ , $x_i \ne y_i$ ) denoting the endpoints of the $i$ -th edge.

It is guaranteed that the given graph is a tree.

Print $n$ pairs of integers, the $i$ -th pair should contain two integers $l_i$ and $r_i$ ( $1 \le l_i < r_i \le 2n$ ) — the endpoints of the $i$ -th segment. All $2n$ integers you print should be unique.

It is guaranteed that the answer always exists.