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 PyPy3 (2.4.0) (judge verdict: Time Limit Exceeded)

import sys
sys.setrecursionlimit(10 ** 9)
n = int(input())
#隣接リスト
adjacent_list = [[] for i in range(n+1)]
#入力されたデータを保存しておくやつ
data = []
#最終的に色を入れておくやつ。
#edgedata[12] => 1->2の辺の色を入れておく
edgedata = {}
for i in range(n-1):
    a, b = list(map(int, input().split()))
    data.append([a,b])
    adjacent_list[a].append(b)
    adjacent_list[b].append(a)
#つながってるノードが一番多いやつを保存しておく
max_node = 0
#探索が終わったやつを入れる
finished = set()
for i in range(1,n):
    if len(adjacent_list[i]) > len(adjacent_list[max_node]):
        max_node = i
#kには色の種類を入れる
k = len(adjacent_list[max_node])
#print("k => {}".format(k))
def dfs(node, num):
    #隣接リストの中身を、つながってるノード順にソート
    node_list = sorted(adjacent_list[node], key = lambda x:len(adjacent_list[x]))
    #さっきつかったいろは使わないようにする
    color_list = [i for i in range(1,k+1) if i != num]
    c_i = 0
    finished.add(node)
    for i in node_list:
        #まだ見てなかったら、dfsを掛ける
        if i not in finished:
            s = min(node, i)
            t = max(node, i)
            #小さいほうから大きいほうに文字を並べ、ハッシュに値を保存
            #print("{}".format(color_list))
            #print("c_i =>{}  node =>{}".format(c_i,node))

            edgedata[(s, t)] = color_list[c_i]
            #color_list[c_i]は使ったので、使った色として引数で渡す。
            dfs(i, color_list[c_i])
            #この色は使ったから、一個ずらす
            c_i += 1
#一番のノードを根とし、dfsをかける
dfs(k,-1)
print(k)
for i in data:
    print(edgedata[tuple(i)])
#output k