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 Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

# 根付き木の原点を任意に定める

n = gets.to_i
edges = []
(n - 1).times { edges << gets.split.map(&:to_i) }
graph = Array.new(n + 1) { Array.new }
# 無向グラフなので両始点のグラフに入れる
edges.each do |f, t|
  graph[f] << [f, t]
  graph[t] << [t, f]
end

ans = Array.new(n + 1) { Hash.new(0) }
queue = [[1, 1, 100001]]
biggest_num = 0
available_list = (1..10 ** 5 + 1).to_a
until queue.empty?
  from, to, num = queue.shift
  tmp_list = available_list.dup
  tmp_list.delete_at(tmp_list.bsearch_index { |i| i >= num })
  graph[to].each do |f, t|
    # 逆流しないようにする（両方のグラフに入れているので必ず1つ該当する）
    next if t == from
    next_num = tmp_list.shift
    ans[f][t] = next_num
    queue << [f, t, next_num]
    biggest_num = next_num if biggest_num < next_num
  end
end

puts biggest_num
edges.each { |f, t| puts ans[f][t] }