Takahashi has an ability to generate a tree using a permutation (p_1,p_2,...,p_n) of (1,2,...,n), in the following process:
First, prepare Vertex 1, Vertex 2, ..., Vertex N.
For each i=1,2,...,n, perform the following operation:
 - If p_i = 1, do nothing.
 - If p_i \neq 1, let j' be the largest j such that p_j < p_i. Span an edge between Vertex i and Vertex j'.
Takahashi is trying to make his favorite tree with this ability.
His favorite tree has n vertices from Vertex 1 through Vertex n, and its i-th edge connects Vertex v_i and w_i.
Determine if he can make a tree isomorphic to his favorite tree by using a proper permutation.
If he can do so, find the lexicographically smallest such permutation.
-----Notes-----
For the definition of isomorphism of trees, see wikipedia. Intuitively, two trees are isomorphic when they are the "same" if we disregard the indices of their vertices.
-----Constraints-----
 - 2 \leq n \leq 10^5
 - 1 \leq v_i, w_i \leq n
 - The given graph is a tree.
-----Input-----
Input is given from Standard Input in the following format:
n
v_1 w_1
v_2 w_2
:
v_{n-1} w_{n-1}
-----Output-----
If there is no permutation that can generate a tree isomorphic to Takahashi's favorite tree, print -1.
If it exists, print the lexicographically smallest such permutation, with spaces in between.
-----Sample Input-----
6
1 2
1 3
1 4
1 5
5 6
-----Sample Output-----
1 2 4 5 3 6
If the permutation (1, 2, 4, 5, 3, 6) is used to generate a tree, it looks as follows:
This is isomorphic to the given graph.
import sys

input=sys.stdin.readline
sys.setrecursionlimit(2*10**5)

n=int(input())
edge=[[] for i in range(n)]
for i in range(n-1):
    v,w=map(int,input().split())
    edge[v-1].append(w-1)
    edge[w-1].append(v-1)

if n==2:
    print(1,2);return

leafcnt=[0]*n
for v in range(n):
    for nv in edge[v]:
        if len(edge[nv])==1:
            leafcnt[v]+=1

used=[False]*n
line=[]
def line_check(v):
    used[v]=True
    line.append(leafcnt[v])
    flag=False
    for nv in edge[v]:
        if not used[nv] and len(edge[nv])!=1:
            if not flag:
                line_check(nv)
                flag=True
            else:
                return False
    return True

for v in range(n):
    if not used[v] and len(edge[v])-leafcnt[v]<=1 and len(edge[v])!=1:
        if not line:
            check=line_check(v)
            if not check:
                print(-1);return()
        else:
            print(-1);return()


line_rev=line[::-1]
res=min(line,line_rev)
res=[0]+res+[0]
res[1]-=1
res[-2]-=1

ans=[]
L=1
for val in res:
    R=L+val
    for i in range(L+1,R+1):
        ans.append(i)
    ans.append(L)
    L=R+1

print(*ans)

import sys

input=sys.stdin.readline
sys.setrecursionlimit(2*10**5)

n=int(input())
edge=[[] for i in range(n)]
for i in range(n-1):
    v,w=map(int,input().split())
    edge[v-1].append(w-1)
    edge[w-1].append(v-1)

if n==2:
    print(1,2);return

leafcnt=[0]*n
for v in range(n):
    for nv in edge[v]:
        if len(edge[nv])==1:
            leafcnt[v]+=1

used=[False]*n
line=[]
def line_check(v):
    used[v]=True
    line.append(leafcnt[v])
    flag=False
    for nv in edge[v]:
        if not used[nv] and len(edge[nv])!=1:
            if not flag:
                line_check(nv)
                flag=True
            else:
                return False
    return True

for v in range(n):
    if not used[v] and len(edge[v])-leafcnt[v]<=1 and len(edge[v])!=1:
        if not line:
            check=line_check(v)
            if not check:
                print(-1);return()
        else:
            print(-1);return()


line_rev=line[::-1]
res=min(line,line_rev)
res=[0]+res+[0]
res[1]-=1
res[-2]-=1

ans=[]
L=1
for val in res:
    R=L+val
    for i in range(L+1,R+1):
        ans.append(i)
    ans.append(L)
    L=R+1

print(*ans)
