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: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Text;
using System.Linq;
namespace Practice
{
    class Program
    {
        static void Main(string[] args)
        {
            var sw = new System.IO.StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
            Console.SetOut(sw);
            Solve();

            Console.Out.Flush();
        }
        private const int MOD = 1000000007;
        private const int MAX = 1000001;
        static long[] fac = new long[MAX];
        static long[] finv = new long[MAX];
        static long[] inv = new long[MAX];
       

        private static int[] visited;
        private static Dictionary<Tuple<int, int>, int> _colors = new Dictionary<Tuple<int, int>, int>();
        private static HashSet<int>[] _usedColors;
        static void Solve()
        {
            var n = Input.NextInt();
            var g = new List<int>[n];
            visited = new int[n];
            _usedColors = new HashSet<int>[n];
            for(int i = 0;i<n;++i)
            {
                _usedColors[i] = new HashSet<int>();
            }
            for(int i=0;i<g.Length;++i)
            {
                g[i] = new List<int>();
            }
            var ai = new int[n - 1];
            var bi = new int[n - 1];
            for(int i=0;i<n-1;++i)
            {
                var a = Input.NextInt()-1;
                var b = Input.NextInt()-1;
                ai[i] = a;
                bi[i] = b;

                g[a].Add(b);
                g[b].Add(a);
            }

            int colorTotal = 0;
            var id_numList = new List<Tuple<int, int>>();

            for(int i=0;i<g.Length;++i)
            {
                var list = g[i];
                colorTotal = Math.Max(list.Count, colorTotal);
                id_numList.Add(new Tuple<int, int>(i, list.Count));
            }

            id_numList.Sort(Compare);
            id_numList.Reverse();


            bfs(g, 0, colorTotal);



            Console.WriteLine(colorTotal);
            for(int i=0;i<n-1;++i)
            {
                var key = new Tuple<int, int>(ai[i], bi[i]);
                int c =_colors[key]; ;
                Console.WriteLine(c);
            }

        }

        private static Queue<int> q = new Queue<int>();
        private static void bfs(List<int>[] g, int s, int maxColor)
        {
            q.Enqueue(s);
            while(q.Count >= 1)
            {
                int u = q.Dequeue();

                var list = g[u];
                foreach(var  v in list)
                {
                    if(visited[v] == 1) { continue; }
                    visited[v] = 1;

                    var key0 = new Tuple<int, int>(u, v);
                    var key1 = new Tuple<int, int>(v, u);

                    if (!_colors.ContainsKey(key0))
                    {
                        // 塗るぞ
                        int color = FindColor(u, v, maxColor);

                        _colors.Add(key0, color);
                        _colors.Add(key1, color);
                        _usedColors[u].Add(color);
                        _usedColors[v].Add(color);
                    }

                    q.Enqueue(v);
                }
            }
        }
        private static int FindColor(int u, int v, int maxColor)
        {
            for(int i=maxColor-1;i>=0;--i)
            {
                int c = i + 1;
                if( !_usedColors[u].Contains(c) && !_usedColors[v].Contains(c))
                {
                    return c;
                }
            }
            return 0;
        }
        private static int Compare(Tuple<int, int> a, Tuple<int, int> b)
        {
            return a.Item2.CompareTo(b.Item2);
        }
        #region よく使う便利関数

        private static Dictionary<long, int> Factorize(long n)
        {
            var dic = new Dictionary<long, int>();

            for (long i = 2; i * i <= n; ++i)
            {
                while (n % i == 0)
                {
                    if (!dic.ContainsKey(i))
                    {
                        dic.Add(i, 0);
                    }
                    dic[i]++;
                    n /= i;
                }

            }

            if (n != 1)
            {
                dic.Add(n, 1);
            }
            return dic;
        }
        static void COMinit()
        {
            fac[0] = fac[1] = 1;
            finv[0] = finv[1] = 1;
            inv[1] = 1;
            for (int i = 2; i < MAX; i++)
            {
                fac[i] = fac[i - 1] * i % MOD;
                inv[i] = MOD - inv[MOD % i] * (MOD / i) % MOD;
                finv[i] = finv[i - 1] * inv[i] % MOD;
            }
        }

        // 二項係数計算
        static long COM(long n, long k)
        {
            if (n < k) return 0;
            if (n < 0 || k < 0) return 0;
            return fac[n] * (finv[k] * finv[n - k] % MOD) % MOD;
        }

        static class Input
        {
            static char[] separator = { ' ' };
            public static bool IsEof { get; set; }
            static Queue<string> q { get; set; }

            static Input()
            {
                IsEof = false;
                q = new Queue<string>();
            }

            /// <summary>
            /// 入力予約
            /// </summary>
            /// <param name="items"></param>
            public static void SetText(IEnumerable<string> items)
            {
                foreach (var item in items)
                {
                    SetText(item);
                }
            }

            /// <summary>
            /// 入力予約
            /// </summary>
            /// <param name="s"></param>
            /// <returns></returns>
            public static bool SetText(string s)
            {
                if (s == null) return false;

                foreach (var elem in s.Trim().Split(separator, StringSplitOptions.RemoveEmptyEntries))
                {
                    q.Enqueue(elem);
                }

                return true;
            }

            /// <summary>
            /// 内部queueに入力からの値をsplitして格納する
            /// </summary>
            /// <returns></returns>
            static bool read()
            {
                var s = Console.ReadLine();
                if (s == null) return false;

                foreach (var elem in s.Trim().Split(separator, StringSplitOptions.RemoveEmptyEntries))
                {
                    q.Enqueue(elem);
                }

                if (!q.Any()) return read();

                return true;
            }

            /// <summary>
            /// 次のstringを一つ読み込む
            /// </summary>
            /// <returns></returns>
            public static string Next()
            {
                if (!q.Any())
                {
                    if (!read())
                    {
                        IsEof = true;
                        return "";
                    }
                }

                return q.Dequeue();
            }

            public static int NextInt() => int.Parse(Next());
            public static long NextLong() => long.Parse(Next());
            public static double NextDouble() => double.Parse(Next());

            public static List<string> Next(int n) => Enumerable.Range(0, n).Select(_ => Next()).ToList();
            public static List<int> NextInt(int n) => Next(n).Select(x => int.Parse(x)).ToList();
            public static List<long> NextLong(int n) => Next(n).Select(x => long.Parse(x)).ToList();
            public static List<double> NextDouble(int n) => Next(n).Select(x => double.Parse(x)).ToList();

            public static List<string> NextLine() => Console.ReadLine().Trim().Split(separator, StringSplitOptions.RemoveEmptyEntries).ToList();
            public static List<int> NextIntLine() => NextLine().Select(x => int.Parse(x)).ToList();
            public static List<long> NextLongLine() => NextLine().Select(x => long.Parse(x)).ToList();
            public static List<double> NextDoubleLine() => NextLine().Select(x => double.Parse(x)).ToList();
        }
        private static bool isPrime(long x)
        {
            if (x == 2) { return true; }
            if (x < 2 || x % 2 == 0) { return false; }
            long i = 3;
            while (i * i <= x)
            {
                if (x % i == 0) { return false; }
                i = i + 2;
            }
            return true;
        }
        private static long lcm(long x, long y)
        {
            return x / gcd(x, y) * y;
        }
        private static long gcd(long x, long y)
        {
            return y > 0 ? gcd(y, x % y) : x;
        }
        private static long pow(long x, long n)
        {
            if (n == 0) { return 1; }
            long res = pow(x * x % MOD, n / 2);
            if (n % 2 == 1)
            {
                res = res * x % MOD;
            }
            return res;
        }
        private static void Swap<T>(ref T x, ref T y)
        {
            T t = x;
            x = y;
            y = t;
        }

        /// <summary>
        /// 指定した値以上の先頭のインデクスを返す
        /// </summary>
        /// <typeparam name="T">比較する値の型</typeparam>
        /// <param name="arr">対象の配列（※ソート済みであること）</param>
        /// <param name="start">開始インデクス [inclusive]</param>
        /// <param name="end">終了インデクス [exclusive]</param>
        /// <param name="value">検索する値</param>
        /// <param name="comparer">比較関数(インターフェイス)</param>
        /// <returns>指定した値以上の先頭のインデクス</returns>
        public static int LowerBound<T>(IReadOnlyList<T> arr, int start, int end, T value, IComparer<T> comparer)
        {
            int low = start;
            int high = end;
            int mid;
            while (low < high)
            {
                mid = ((high - low) >> 1) + low;
                if (comparer.Compare(arr[mid], value) < 0)
                    low = mid + 1;
                else
                    high = mid;
            }
            return low;
        }

        //引数省略のオーバーロード
        public static int LowerBound<T>(IReadOnlyList<T> arr, T value) where T : IComparable
        {
            return LowerBound(arr, 0, arr.Count, value, Comparer<T>.Default);
        }

        /// <summary>
        /// 指定した値より大きい先頭のインデクスを返す
        /// </summary>
        /// <typeparam name="T">比較する値の型</typeparam>
        /// <param name="arr">対象の配列（※ソート済みであること）</param>
        /// <param name="start">開始インデクス [inclusive]</param>
        /// <param name="end">終了インデクス [exclusive]</param>
        /// <param name="value">検索する値</param>
        /// <param name="comparer">比較関数(インターフェイス)</param>
        /// <returns>指定した値より大きい先頭のインデクス</returns>
        public static int UpperBound<T>(IReadOnlyList<T> arr, int start, int end, T value, IComparer<T> comparer)
        {
            int low = start;
            int high = end;
            int mid;
            while (low < high)
            {
                mid = ((high - low) >> 1) + low;
                if (comparer.Compare(arr[mid], value) <= 0)
                    low = mid + 1;
                else
                    high = mid;
            }
            return low;
        }

        //引数省略のオーバーロード
        public static int UpperBound<T>(IReadOnlyList<T> arr, T value)
        {
            return UpperBound(arr, 0, arr.Count, value, Comparer<T>.Default);
        }
#endregion
    }

    public class DisjointSet
    {
        private int[] rank;
        private int[] p;
        private int[] size;
        public DisjointSet(int n)
        {
            rank = new int[n];
            p = new int[n];
            size = new int[n];
            for (int i = 0; i < n; ++i)
            {
                MakeSet(i);
            }
        }
        public void MakeSet(int x)
        {
            p[x] = x;
            rank[x] = 0;
            size[x] = 1;
        }
        public bool IsSame(int x, int y)
        {
            return FindSet(x) == FindSet(y);
        }
        public void Unite(int x, int y)
        {
            Link(FindSet(x), FindSet(y));
        }
        public void Link(int x, int y)
        {
            if (rank[x] > rank[y])
            {
                p[y] = x;
                size[x] += size[y];
                size[y] = 0;
            }
            else
            {
                p[x] = y;
                size[y] += size[x];
                size[x] = 0;
                if (rank[x] == rank[y])
                {
                    rank[y]++;
                }
            }
        }
        public int FindSet(int x)
        {
            if (x != p[x])
            {
                p[x] = FindSet(p[x]);
            }
            return p[x];
        }
        public int Size(int x)
        {
            int root = FindSet(x);
            return size[root];
        }
    }
    public class PriorityQueue<T> where T : IComparable
    {
        private IComparer<T> _comparer = null;
        private int _type = 0;

        private T[] _heap;
        private int _sz = 0;

        private int _count = 0;

        /// <summary>
        /// Priority Queue with custom comparer
        /// </summary>
        public PriorityQueue(int maxSize, IComparer<T> comparer)
        {
            _heap = new T[maxSize];
            _comparer = comparer;
        }

        /// <summary>
        /// Priority queue
        /// </summary>
        /// <param name="maxSize">max size</param>
        /// <param name="type">0: asc, 1:desc</param>
        public PriorityQueue(int maxSize, int type = 0)
        {
            _heap = new T[maxSize];
            _type = type;
        }

        private int Compare(T x, T y)
        {
            if (_comparer != null) return _comparer.Compare(x, y);
            return _type == 0 ? x.CompareTo(y) : y.CompareTo(x);
        }

        public void Push(T x)
        {
            _count++;

            //node number
            var i = _sz++;

            while (i > 0)
            {
                //parent node number
                var p = (i - 1) / 2;

                if (Compare(_heap[p], x) <= 0) break;

                _heap[i] = _heap[p];
                i = p;
            }

            _heap[i] = x;
        }

        public T Pop()
        {
            _count--;

            T ret = _heap[0];
            T x = _heap[--_sz];

            int i = 0;
            while (i * 2 + 1 < _sz)
            {
                //children
                int a = i * 2 + 1;
                int b = i * 2 + 2;

                if (b < _sz && Compare(_heap[b], _heap[a]) < 0) a = b;

                if (Compare(_heap[a], x) >= 0) break;

                _heap[i] = _heap[a];
                i = a;
            }

            _heap[i] = x;

            return ret;
        }

        public int Count()
        {
            return _count;
        }

        public T Peek()
        {
            return _heap[0];
        }

        public bool Contains(T x)
        {
            for (int i = 0; i < _sz; i++) if (x.Equals(_heap[i])) return true;
            return false;
        }

        public void Clear()
        {
            while (this.Count() > 0) this.Pop();
        }

        public IEnumerator<T> GetEnumerator()
        {
            var ret = new List<T>();

            while (this.Count() > 0)
            {
                ret.Add(this.Pop());
            }

            foreach (var r in ret)
            {
                this.Push(r);
                yield return r;
            }
        }

        public T[] ToArray()
        {
            T[] array = new T[_sz];
            int i = 0;

            foreach (var r in this)
            {
                array[i++] = r;
            }

            return array;
        }
    }
}