Score : $600$ points

Problem Statement

We have a tree $T$ with vertices numbered $1$ to $N$. The $i$-th edge of $T$ connects vertex $u_i$ and vertex $v_i$.

Let us use $T$ to define the similarity of a permutation $P = (P_1,P_2,\ldots,P_N)$ of $(1,2,\ldots,N)$ as follows.

• For a simple path $x=(x_1,x_2,\ldots,x_k)$ in $T$, let $y=(P_{x_1}, P_{x_2},\ldots,P_{x_k})$. The similarity is the maximum possible length of a longest common subsequence of $x$ and $y$.

Construct a permutation $P$ with the minimum similarity.

Constraints

• $2 \leq N \leq 5000$
• $1\leq u_i,v_i\leq N$
• The given graph is a tree.
• All numbers in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$

$u_1$ $v_1$

$u_2$ $v_2$

$\vdots$

$u_{N-1}$ $v_{N-1}$

Output

Print a permutation $P$ with the minimum similarity, separated by spaces. If multiple solutions exist, you may print any of them.

3
1 2
2 3

3 2 1

This permutation has a similarity of $1$, which can be computed as follows.

• For $x=(1)$, we have $y=(P_1)=(3)$. The length of a longest common subsequence of $x$ and $y$ is $0$.
• For $x=(2)$, we have $y=(P_2)=(2)$. The length of a longest common subsequence of $x$ and $y$ is $1$.
• For $x=(3)$, we have $y=(P_2)=(1)$. The length of a longest common subsequence of $x$ and $y$ is $0$.
• For $x=(1,2)$, we have $y=(P_1,P_2)=(3,2)$. The length of a longest common subsequence of $x$ and $y$ is $1$. The same goes for $x=(2,1)$, the reversal of $(1,2)$.
• For $x=(2,3)$, we have $y=(P_2,P_3)=(2,1)$. The length of a longest common subsequence of $x$ and $y$ is $1$. The same goes for $x=(3,2)$, the reversal of $(2,3)$.
• For $x=(1,2,3)$, we have $y=(P_1,P_2,P_3)=(3,2,1)$. The length of a longest common subsequence of $x$ and $y$ is $1$. The same goes for $x=(3,2,1)$, the reversal of $(1,2,3)$.

We can prove that no permutation has a similarity of $0$ or less, so this permutation is a valid answer.

4
2 1
2 3
2 4

3 4 1 2

If multiple permutations have the minimum similarity, you may print any of them. For instance, you may also print 4 3 2 1.