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 OCaml (4.02.3) (judge verdict: Accepted)

type edge = {a: int; b: int; i: int}

let degrees edges =
  let n = (List.length edges) + 1 in
  let rec helper edges acc =
    match edges with
    | [] -> acc
    | {a=a; b=b} :: rst ->
      acc.(a) <- acc.(a) + 1;
      acc.(b) <- acc.(b) + 1;
      helper rst acc in
  helper edges (Array.make n 0)
;;

let solve n edges =
  let k =
    let rec find_max l i m =
      if i == Array.length l then m
      else if l.(i) > m then find_max l (i+1) l.(i)
      else find_max l (i+1) m in
    find_max (degrees edges) 0 0 in
  let p = Array.make n 0 in
  let cs = Array.make (n-1) 0 in
  let rec choose_color es prev cnt =
    match es with
    | [] -> cs
    | e :: rst ->
      let cnt = if e.a == prev then cnt else 0 in
      let color = if e.a == 0 then cnt
        else if cnt < p.(e.a) then cnt else cnt + 1 in
      p.(e.b) <- color; cs.(e.i) <- color;
      choose_color rst e.a (cnt+1) in
  (k, choose_color edges 0 0)
;;

let read () =
  let n = read_int () in
  let rec read_edges i acc =
    if i == n-1 then acc else
      let input = Str.split (Str.regexp " ") (read_line ()) in
      let a::b::[] = List.map int_of_string input in
      let edge = {a=a-1; b=b-1; i=i} in
      read_edges (i+1) (edge :: acc) in
  let edges = List.sort (fun e1 e2 -> e1.a - e2.a) (read_edges 0 []) in
  (n, edges) in
let print k colors =
  print_endline (string_of_int k);
  for i = 0 to ((Array.length colors) - 1) do
    print_endline (string_of_int (colors.(i) + 1))
  done in
let n, edges = read() in
let k, colors = solve n edges in
print k colors