Problem:
Score : 400 points

Problem Statement

Given is a tree G with N vertices.
The vertices are numbered 1 through N, and the i-th edge connects Vertex a_i and Vertex b_i.

Consider painting the edges in G with some number of colors.
We want to paint them so that, for each vertex, the colors of the edges incident to that vertex are all different.

Among the colorings satisfying the condition above, construct one that uses the minimum number of colors.

Constraints

2 \le N \le 10^5

1 \le a_i \lt b_i \le N

All values in input are integers.

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N
a_1 b_1
a_2 b_2
\vdots
a_{N-1} b_{N-1}

Output

Print N lines.

The first line should contain K, the number of colors used.

The (i+1)-th line (1 \le i \le N-1) should contain c_i, the integer representing the color of the i-th edge, where 1 \le c_i \le K must hold.

If there are multiple colorings with the minimum number of colors that satisfy the condition, printing any of them will be accepted.

Sample Input 1

3
1 2
2 3

Sample Output 1

2
1
2

Sample Input 2

8
1 2
2 3
2 4
2 5
4 7
5 6
6 8

Sample Output 2

4
1
2
3
4
1
1
2

Sample Input 3

6
1 2
1 3
1 4
1 5
1 6

Sample Output 3

5
1
2
3
4
5

Solution in Python (3.8.2) (judge verdict: Time Limit Exceeded)

#!/usr/bin/env python3
import sys
sys.setrecursionlimit(10**7)
import bisect
import heapq
import itertools
import math
import numpy as np
from collections import Counter, defaultdict, deque
from copy import deepcopy
from decimal import Decimal
from math import gcd
from operator import add, itemgetter, mul, xor
def cmb(n,r,mod):
  bunshi=1
  bunbo=1
  for i in range(r):
    bunbo = bunbo*(i+1)%mod
    bunshi = bunshi*(n-i)%mod
  return (bunshi*pow(bunbo,mod-2,mod))%mod
mod = 10**9+7
def I(): return int(input())
def LI(): return list(map(int,input().split()))
def MI(): return map(int,input().split())
def LLI(n): return [list(map(int, input().split())) for _ in range(n)]

n = I()
#edge[i][j]には頂点iと頂点jをつなぐ辺の色を格納する
edge = [[0]*(n+1) for _ in range(n+1)]
#graph[i]には頂点iと繋がっている頂点を格納する
graph = [[] for _ in range(n+1)]
k = 0
a_b = []
for i in range(n-1):
    a,b = MI()
    a_b.append([a,b])
    graph[a].append(b)
    graph[b].append(a)
#check[i] == -1ならば未探索
check = [-1]*(n+1)
check[0] = 0
check[1] = 0
d = deque()
d.append(1)
while d:
    v = d.popleft()
    check[v] = 1
    cnt = 0
    for i in graph[v]:
      if check[i] != -1:
        continue
      cnt = cnt+1
      while cnt in edge[v]:
        cnt = cnt + 1
      edge[i][v] = cnt
      edge[v][i] = cnt
      d.append(i)
      check[i] = 0
      k = max(k,cnt)
print(k)
for i in range(n-1):
    color = edge[a_b[i][0]][a_b[i][1]]
    print(color)