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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Text.RegularExpressions;

public class ABC
{
    //          long[] sp = Console.ReadLine().Split().Select(long .Parse).ToArray();
    //          int[] sp = Console.ReadLine().Split().Select(int.Parse).ToArray();
    //          int N =  int.Parse(Console.ReadLine());
    //CompLib.Collections.PriorityQueue<?>

    public static void Main()
    {
        int N = int.Parse(Console.ReadLine());

        Tuple<int, int>[] tuples = new Tuple<int, int>[N-1];

        int[] chosen = new int[N];
        vertices[] vertices = new vertices[N+1];

        for (int i = 1; i < N+1; i++)
        {
            vertices[i] = new vertices(i);
        }

        for (int i = 0; i < N-1; i++)
        {
            int[] sp = Console.ReadLine().Split().Select(int.Parse).ToArray();
            tuples[i] = new Tuple<int, int>(sp[0], sp[1]);

            chosen[sp[0] - 1]++;
            chosen[sp[1] - 1]++;

            vertices[sp[0]].next.Add(sp[1]);
            vertices[sp[1]].next.Add(sp[0]);

        }
        int max = chosen.Max();
        int maxindex = -1;
        for (int i = 0; i < N-1; i++)
        {
            if (chosen[i] == max) maxindex = i + 1;
        }
        Dictionary<Tuple<int, int>, int> node_color = new Dictionary<Tuple<int, int>, int>();

        int color = 1;

        bool[] finishedColor = new bool[N + 1];
        finishedColor[0] = true;
        Queue<chouten> tasks = new Queue<chouten>();

        for (int i = 0; i < tuples.Length; i++)
        {
            node_color[tuples[i]] = 0;

            if (tuples[i].Item1 == maxindex || tuples[i].Item2 == maxindex)
            {
                int targetnode = 0;
                if (tuples[i].Item1 == maxindex) targetnode = tuples[i].Item2;
                else targetnode = tuples[i].Item1;

                node_color[tuples[i]] = color;
                chouten chouten = new chouten(tuples[i], color, targetnode);
                tasks.Enqueue(chouten);

                finishedColor[maxindex] = true;

                color++;
            }

        }



        while (tasks.Count > 0)
        {
            chouten thistask = tasks.Dequeue();
            if (finishedColor[thistask.targetChouten]) continue;

            color = 1;
            if (thistask.nuttaColor == color) color = 2;

            for (int i = 0; i < vertices[thistask.targetChouten].next.Count; i++)
            {
                int target = vertices[thistask.targetChouten].next[i];
                int bigger = Math.Max(target, thistask.targetChouten);
                int smaller = Math.Min(target, thistask.targetChouten);
                Tuple<int, int> node = new Tuple<int, int>(smaller, bigger);
                if (node_color[node] != 0)
                {


                }
                else
                {
                    node_color[node] = color;
                    tasks.Enqueue(new chouten(node, color, target));

                    color++;
                    if (thistask.nuttaColor == color) color++;
                }

            }
            //今の頂点
            finishedColor[thistask.targetChouten] = true;


        }


        //input 塗った頂点同士、その色、対象とする頂点

        //塗った結果を格納し、対象とする頂点を

        Console.WriteLine(max);
        for (int i = 0; i < N-1; i++)
        {
            Console.WriteLine(node_color[tuples[i]]);
        }

    }

    


    public static int comparer(Tuple<int, int> a, Tuple<int, int> b)
    {
        if (a.Item1 > b.Item1) return 1;
        else if (a.Item1 < b.Item1) return -1;
        else if (a.Item2 > b.Item2) return -1;
        else if (a.Item2 < b.Item2) return 1;
        else return 0;
    }
}

class vertices
{
    public int index;
    public List<int> next;

    public vertices()
    {

    }
    public vertices(int index)
    {
        this.index = index;
        this.next = new List<int>();
    }
}

class chouten
{
    public Tuple<int, int> nuttaNode;
    public int nuttaColor;
    public int targetChouten;

    public chouten()
    {
    }

    public chouten(Tuple<int, int> nuttnaode, int color, int target)
    {
        this.nuttaNode = nuttnaode;
        this.nuttaColor = color;
        this.targetChouten = target;
    }

}

class readint
{



    public readint(out int i)
    {
        i = int.Parse(Console.ReadLine());

    }
    public readint(out int a, out int b)
    {
        int[] sp = Console.ReadLine().Split().Select(int.Parse).ToArray();
        a = sp[0];
        b = sp[1];

    }
    public readint(out int a, out int b, out int c)
    {
        int[] sp = Console.ReadLine().Split().Select(int.Parse).ToArray();
        a = sp[0];
        b = sp[1];
        c = sp[2];
    }
    public readint(out int a, out int b, out int c, out int d)
    {
        int[] sp = Console.ReadLine().Split().Select(int.Parse).ToArray();
        a = sp[0];
        b = sp[1];
        c = sp[2];
        d = sp[3];

    }

}

class readlong
{


    public readlong(out long i)
    {
        i = long.Parse(Console.ReadLine());

    }
    public readlong(out long a, out long b)
    {
        long[] sp = Console.ReadLine().Split().Select(long.Parse).ToArray();
        a = sp[0];
        b = sp[1];

    }
    public readlong(out long a, out long b, out long c)
    {
        long[] sp = Console.ReadLine().Split().Select(long.Parse).ToArray();
        a = sp[0];
        b = sp[1];
        c = sp[2];
    }
    public readlong(out long a, out long b, out long c, out long d)
    {
        long[] sp = Console.ReadLine().Split().Select(long.Parse).ToArray();
        a = sp[0];
        b = sp[1];
        c = sp[2];
        d = sp[3];

    }

}







class Util
{





    public static long GCD(long a, long b)
    {
        if (a < b)
            swap(ref a, ref b);
        if (a % b == 0) return b;
        return GCD(b, a % b);

    }
    public static int GCD(int a, int b)
    {
        if (a < b)
            swap(ref a, ref b);
        if (a % b == 0) return b;
        return GCD(b, a % b);
    }

    public static void swap<T>(ref T a, ref T b)
    {
        T temp = b;
        b = a;
        a = temp;

    }
    public static long LCM(long a, long b)
    {
        return a * b / GCD(a, b);
    }
    public static long LCM(int a, int b)
    {
        return Math.BigMul(a, b) / GCD(a, b);
    }


    public static int M = (int)(Math.Pow(10, 9)) + 7;
    private static int[] factorial_modM;
    public static int Multiple_modM(int a, int b)
    {
        return (int)(Math.BigMul(a, b) % M);
    }
    public static void factorial_modM_Prepare(int n)
    {
        factorial_modM = new int[n + 1];
        factorial_modM[0] = 1;
        for (int i = 1; i <= n; ++i)
        {
            factorial_modM[i] = Multiple_modM(factorial_modM[i - 1], i);
        }
    }
    public static int Factorial(int n)
    {
        return factorial_modM[n];
    }
    public static int Pow(int a, int m)
    {
        switch (m)
        {
            case 0:
                return 1;
            case 1:
                return a;
            default:
                int p1 = Pow(a, m / 2);
                int p2 = Multiple_modM(p1, p1);
                return ((m % 2) == 0) ? p2 : Multiple_modM(p2, a);
        }
    }
    public static int Div(int a, int b)
    {
        return Multiple_modM(a, Pow(b, M - 2));
    }
    public static int nCr_modM(int n, int r)
    {
        if (r < 0) return 0;

        if (n < r) { return 0; }
        if (n == r) { return 1; }
        int res = Factorial(n);
        res = Div(res, Factorial(r));
        res = Div(res, Factorial(n - r));
        return res;
    }

}

class UnionFind<T>
{
    public int tree_height;
    public UnionFind<T> parent;
    public T item
    {
        get;
        private set;
    }

    public UnionFind(T item)
    {
        tree_height = 0;
        parent = this;
    }

    public UnionFind<T> FindAdam()
    {
        if (parent == this) return this;
        else
        {
            UnionFind<T> ParentOfParent = parent.FindAdam();
            parent = ParentOfParent;//縦長な構成をつなぎ直している
            return ParentOfParent;
        }
    }
    public static void Unite(UnionFind<T> a, UnionFind<T> b)
    {
        UnionFind<T> ParentOfA = a.FindAdam();
        UnionFind<T> ParentOfB = b.FindAdam();
        if (ParentOfA.tree_height < ParentOfB.tree_height)
        {
            ParentOfA.parent = ParentOfB;
            ParentOfB.tree_height = Math.Max(ParentOfA.tree_height + 1, ParentOfB.tree_height);
        }
        else
        {
            ParentOfB.parent = ParentOfA;
            ParentOfA.tree_height = Math.Max(ParentOfB.tree_height + 1, ParentOfA.tree_height);
        }

    }

}

//Treap 平衡二分木
class Treap<T> where T : IComparable
{
    private class Node
    {
        private static Random g_rand = new Random();
        private readonly int m_rank = g_rand.Next();
        private readonly T m_value;
        private Node m_lch;
        private Node m_rch;
        private int m_count;

        public Node(T value)
        {
            m_value = value;
            m_count = 1;
        }

        private static int Count(Node node)
        {
            return (node != null) ? node.m_count : 0;
        }

        private Node Update()
        {
            m_count = Count(m_lch) + Count(m_rch) + 1;
            return this;
        }

        public static Node Merge(Node a, Node b)
        {
            if (a == null) { return b; }
            if (b == null) { return a; }

            if (a.m_rank < b.m_rank)
            {
                a.m_rch = Merge(a.m_rch, b);
                return a.Update();
            }
            else
            {
                b.m_lch = Merge(a, b.m_lch);
                return b.Update();
            }
        }

        public static Tuple<Node, Node> Split(Node t, int k)
        {
            if (t == null) { return new Tuple<Node, Node>(null, null); }

            if (k <= Count(t.m_lch))
            {
                var pair = Split(t.m_lch, k);
                t.m_lch = pair.Item2;
                return new Tuple<Node, Node>(pair.Item1, t.Update());
            }
            else
            {
                var pair = Split(t.m_rch, k - Count(t.m_lch) - 1);
                t.m_rch = pair.Item1;
                return new Tuple<Node, Node>(t.Update(), pair.Item2);
            }
        }

        public int FindIndex(T value)
        {
            int L = Count(m_lch);
            if (value.CompareTo(m_value) < 0)
            {
                return (m_lch != null) ? m_lch.FindIndex(value) : 0;
            }
            else if (value.CompareTo(m_value) > 0)
            {
                return (m_rch != null) ? m_rch.FindIndex(value) + L + 1 : L + 1;
            }
            else
            {
                return L;
            }
        }

        public T this[int i]
        {
            get
            {
                int L = Count(m_lch);
                if (i < L)
                {
                    return m_lch[i];
                }
                else if (i > L)
                {
                    return m_rch[i - L - 1];
                }
                else
                {
                    return m_value;
                }
            }
        }
    }

    private Node node;

    public void Insert(T value)
    {
        if (node != null)
        {
            int k = node.FindIndex(value);
            var pair = Node.Split(node, k);
            node = Node.Merge(Node.Merge(pair.Item1, new Node(value)), pair.Item2);
        }
        else
        {
            node = new Node(value);
        }
    }

    public int FindIndex(T value)
    {
        return node.FindIndex(value);
    }

    public T this[int i]
    {
        get
        {
            return node[i];
        }
    }
}

static class Permutation<T>
{
    private static void Search(List<T[]> perms, Stack<T> stack, T[] a)
    {
        int N = a.Length;
        if (N == 0)
        {
            perms.Add(stack.Reverse().ToArray());
        }
        else
        {
            var b = new T[N - 1];
            Array.Copy(a, 1, b, 0, N - 1);
            for (int i = 0; i < a.Length; ++i)
            {
                stack.Push(a[i]);
                Search(perms, stack, b);
                if (i < b.Length) { b[i] = a[i]; }
                stack.Pop();
            }
        }
    }
    public static IEnumerable<T[]> All(IEnumerable<T> src)
    {
        var perms = new List<T[]>();
        Search(perms, new Stack<T>(), src.ToArray());
        return perms;
    }
}

namespace CompLib.Collections
{
    #region PriorityQueue
    /// <summary>
    /// 指定した型のインスタンスを最も価値が低い順に取り出すことが可能な可変サイズのコレクションを表します．
    /// </summary>
    /// <typeparam name="T">優先度付きキュー内の要素の型を指定します．</typeparam>
    /// <remarks>内部的にはバイナリヒープによって実装されています．</remarks>
    public class PriorityQueue<T>
    {
        readonly List<T> heap = new List<T>();
        readonly Comparison<T> cmp;

        /// <summary>
        /// デフォルトの比較子を使用してインスタンスを初期化します．
        /// </summary>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public PriorityQueue() { cmp = Comparer<T>.Default.Compare; }

        /// <summary>
        /// デリゲートで表されるような比較関数を使用してインスタンスを初期化します．
        /// </summary>
        /// <param name="comparison"></param>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public PriorityQueue(Comparison<T> comparison) { cmp = comparison; }

        /// <summary>
        /// 指定された比較子を使用してインスタンスを初期化します．
        /// </summary>
        /// <param name="comparer"></param>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public PriorityQueue(IComparer<T> comparer) { cmp = comparer.Compare; }

        /// <summary>
        /// 優先度付きキューに要素を追加します．
        /// </summary>
        /// <param name="item">優先度付きキューに追加される要素</param>
        /// <remarks>最悪計算量 O(log N) で実行されます．</remarks>
        public void Enqueue(T item)
        {
            var pos = heap.Count;
            heap.Add(item);
            while (pos > 0)
            {
                var par = (pos - 1) / 2;
                if (cmp(heap[par], item) <= 0)
                    break;
                heap[pos] = heap[par];
                pos = par;
            }
            heap[pos] = item;

        }

        /// <summary>
        /// 優先度付きキューから最も価値が低い要素を削除し，返します．
        /// </summary>
        /// <returns>優先度付きキューから削除された要素．</returns>
        /// <remarks>最悪計算量 O(log N) で実行されます．</remarks>
        public T Dequeue()
        {
            var ret = heap[0];
            var pos = 0;
            var x = heap[heap.Count - 1];

            while (pos * 2 + 1 < heap.Count - 1)
            {
                var lch = pos * 2 + 1;
                var rch = pos * 2 + 2;
                if (rch < heap.Count - 1 && cmp(heap[rch], heap[lch]) < 0) lch = rch;
                if (cmp(heap[lch], x) >= 0)
                    break;
                heap[pos] = heap[lch];
                pos = lch;
            }
            heap[pos] = x;
            heap.RemoveAt(heap.Count - 1);
            return ret;

        }
        /// <summary>
        ///  優先度付きキューに含まれる最も価値が低い要素を削除せずに返します．
        /// </summary>
        /// <returns>優先度付きキューに含まれる最も価値が低い要素．</returns>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public T Peek() { return heap[0]; }

        /// <summary>
        /// 優先度付きキュー内の要素の数を取得します．
        /// </summary>
        /// <returns>優先度付キュー内にある要素の数</returns>
        /// <remarks>最悪計算量 O(1) で実行されます．</remarks>
        public int Count { get { return heap.Count; } }
        /// <summary>
        /// 優先度付きキュー内に要素が存在するかどうかを O(1) で判定します．
        /// </summary>
        /// <returns>優先度付キュー内にある要素が存在するならば true，そうでなければ　false．</returns>
        /// <remarks>この操作は O(1) で実行されます．</remarks>
        public bool Any() { return heap.Count > 0; }

        /// <summary>
        /// 優先度付きキューに含まれる要素を昇順に並べて返します．
        /// </summary>
        /// <remarks>この操作は計算量 O(N log N)で実行されます．</remarks>
        public T[] Items
        {
            get
            {
                var ret = heap.ToArray();
                Array.Sort(ret, cmp);
                return ret;
            }
        }
    }
    #endregion
}

/// <summary>
/// SEGTREEは1-BASE運用でございます
/// よろしくおねがいします
/// </summary>
/// <typeparam name="T"></typeparam>

class Segtree<T>
{
    int n;//高さ
    T[] tree;//本体
    Func<T, T, T> f;//更新関数。a -> b -> ab使ったなにか
    T exnum;//初期値
    int count;
    public Segtree(int m, Func<T, T, T> f, T ex)
    {
        this.count = 0;
        this.f = f;
        this.exnum = ex;
        n = 1;
        while (n < m) n <<= 1;

        tree = new T[(n << 1) - 1];
        for (int i = 0; i < tree.Length; i++) tree[i] = ex;
    }
    public Segtree(int m, T ini, Func<T, T, T> f, T ex) : this(m, f, ex)
    {
        this.count = 0;
        for (int i = 0; i < m; ++i) tree[i + n - 1] = ini;
        all_update();
    }
    public void assign_without_update(int j, T x)
    {
        tree[j + n - 1] = x;
    }
    public void update(int j, T x)//j番目をxにする
    {
        int i = j + n - 1;
        tree[i] = x;
        while (i > 0)
        {
            i = (i - 1) >> 1;
            tree[i] = f(tree[(i << 1) + 1], tree[(i << 1) + 2]);
        }
        this.count++;
    }


    public void all_update()
    {
        for (int i = n - 2; i >= 0; i--)
            tree[i] = f(tree[(i << 1) + 1], tree[(i << 1) + 2]);
    }
    public T look(int i) { return tree[i + n - 1]; }

    public void delete(int j)
    {
        int i = j + n - 1;
        tree[i] = exnum;
        int moved = 0;

        while (true)
        {
            tree[i + 1] = tree[i];
            T check = tree[i];
            moved++;
            if (moved + j + 2 > count)
            {
                break;
            }

        }
        all_update();
    }


    // [s, t)
    public T run(int s, int t) { return query(s, t, 0, 0, n); }
    T query(int s, int t, int k, int l, int r)
    {
        if (r <= s || t <= l) return exnum;
        if (s <= l && r <= t) return tree[k];

        return f(query(s, t, (k << 1) + 1, l, (l + r) >> 1), query(s, t, (k + 1) << 1, (l + r) >> 1, r));
    }
}