Problem:
Score : 1100 points

Problem Statement

Coloring of the vertices of a tree G is called a good coloring when, for every pair of two vertices u and v painted in the same color, picking u as the root and picking v as the root would result in isomorphic rooted trees.

Also, the colorfulness of G is defined as the minimum possible number of different colors used in a good coloring of G.

You are given a tree with N vertices. The vertices are numbered 1 through N, and the i-th edge connects Vertex a_i and Vertex b_i. We will construct a new tree T by repeating the following operation on this tree some number of times:

Add a new vertex to the tree by connecting it to one of the vertices in the current tree with an edge.

Find the minimum possible colorfulness of T.
Additionally, print the minimum number of leaves (vertices with degree 1) in a tree T that achieves the minimum colorfulness.

Notes

The phrase "picking u as the root and picking v as the root would result in isomorphic rooted trees" for a tree G means that there exists a bijective function f_{uv} from the vertex set of G to itself such that both of the following conditions are met:

f_{uv}(u)=v

For every pair of two vertices (a,b), edge (a,b) exists if and only if edge (f_{uv}(a),f_{uv}(b)) exists.

Constraints

2 \leq N \leq 100

1 \leq a_i,b_i \leq N(1\leq i\leq N-1)

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N
a_1 b_1
:
a_{N-1} b_{N-1}

Output

Print two integers with a space in between.
First, print the minimum possible colorfulness of a tree T that can be constructed.
Second, print the minimum number of leaves in a tree that achieves it.

It can be shown that, under the constraints of this problem, the values that should be printed fit into 64-bit signed integers.

Sample Input 1

5
1 2
2 3
3 4
3 5

Sample Output 1

2 4

If we connect a new vertex 6 to vertex 2, painting the vertices (1,4,5,6) red and painting the vertices (2,3) blue is a good coloring.
Since painting all the vertices in a single color is not a good coloring, we can see that the colorfulness of this tree is 2.
This is actually the optimal solution. There are four leaves, so we should print 2 and 4.

Sample Input 2

8
1 2
2 3
4 3
5 4
6 7
6 8
3 6

Sample Output 2

3 4

Sample Input 3

10
1 2
2 3
3 4
4 5
5 6
6 7
3 8
5 9
3 10

Sample Output 3

4 6

Sample Input 4

13
5 6
6 4
2 8
4 7
8 9
3 2
10 4
11 10
2 4
13 10
1 8
12 1

Sample Output 4

4 12

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

using System;
using System.Collections.Generic;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Linq;
using System.Text;
using System.Threading;

namespace ReadWriteTemplate
{
    public static class Solver
    {
        private static void SolveCase()
        {
            int n = ReadInt();

            var vertices = new Vertex[n];
            for (int i = 0; i < n; i++)
            {
                vertices[i] = new Vertex(i, n);
            }

            for (int i = 0; i < n - 1; i++)
            {
                int x = ReadInt() - 1;
                int y = ReadInt() - 1;
                vertices[x].Edges.Add(vertices[y]);
                vertices[y].Edges.Add(vertices[x]);
            }

            long ansColors = int.MaxValue;
            long ansLists = int.MaxValue;

            for (int i = 0; i < n; i++)
            {
                // vertex
                for (int j = 0; j < n; j++)
                {
                    vertices[j].Visited = false;
                    vertices[j].Depth = 0;
                }
                Vertex.DepthMax = Enumerable.Repeat(1, n + 1).ToArray();

                vertices[i].Dfs(null);

                long colors = vertices.Max(v => v.Depth) + 1;
                long lists = vertices[i].Edges.Count;
                for (int j = 1; j < colors; j++)
                {
                    lists *= Vertex.DepthMax[j];
                }
                if (lists == 0)
                {
                    lists = 1;
                }
                if (ansColors > colors || ansColors == colors && ansLists > lists)
                {
                    ansColors = colors;
                    ansLists = lists;
                }

                // edge
                foreach (Vertex t in vertices[i].Edges)
                {
                    for (int j = 0; j < n; j++)
                    {
                        vertices[j].Visited = false;
                        vertices[j].Depth = 0;
                    }
                    Vertex.DepthMax = Enumerable.Repeat(1, n + 1).ToArray();

                    vertices[i].Dfs(t);
                    t.Dfs(vertices[i]);

                    colors = vertices.Max(v => v.Depth) + 1;
                    lists = 2 * (Math.Max(vertices[i].Edges.Count, t.Edges.Count) - 1);
                    for (int j = 1; j < colors; j++)
                    {
                        lists *= Vertex.DepthMax[j];
                    }
                    if (lists == 0)
                    {
                        lists = 2;
                    }
                    if (ansColors > colors || ansColors == colors && ansLists > lists)
                    {
                        ansColors = colors;
                        ansLists = lists;
                    }
                }
            }

            Writer.WriteLine($"{ansColors} {ansLists}");
        }

        public class Vertex
        {
            public Vertex(int id, int n)
            {
                Id = id;
                Edges = new List<Vertex>();
            }

            public int Id;

            public List<Vertex> Edges;


            public long Depth;

            public bool Visited;

            public static int[] DepthMax;

            public void Dfs(Vertex p)
            {
                Visited = true;
                foreach (Vertex vertex in Edges)
                {
                    if (!vertex.Visited && vertex != p)
                    {
                        vertex.Depth = Depth + 1;
                        vertex.Dfs(p);
                    }
                }
                DepthMax[Depth] = Math.Max(DepthMax[Depth], Edges.Count - 1);
            }
        }

        public static void Solve()
        {
            SolveCase();

            /*var sw = Stopwatch.StartNew();*/

            /*int T = ReadInt();
            for (int i = 0; i < T; i++)
            {
                Writer.Write("Case #{0}: ", i + 1);
                SolveCase();
            }*/

            /*sw.Stop();
            Console.WriteLine(sw.ElapsedMilliseconds);*/
        }

        public static void Main()
        {
            Thread.CurrentThread.CurrentCulture = CultureInfo.InvariantCulture;

#if DEBUG
            //Reader = Console.In; Writer = Console.Out;
            Reader = File.OpenText("input.txt"); Writer = File.CreateText("output.txt");
#else
            Reader = Console.In; Writer = Console.Out;
#endif

            Solve();
            /*Thread thread = new Thread(Solve, 64 * 1024 * 1024);
            thread.CurrentCulture = CultureInfo.InvariantCulture;
            thread.Start();
            thread.Join();*/

            Reader.Close();
            Writer.Close();
        }

        public static IOrderedEnumerable<TSource> OrderByWithShuffle<TSource, TKey>(this IEnumerable<TSource> source, Func<TSource, TKey> keySelector)
        {
            return source.Shuffle().OrderBy(keySelector);
        }

        public static T[] Shuffle<T>(this IEnumerable<T> source)
        {
            T[] result = source.ToArray();
            Random rnd = new Random();
            for (int i = result.Length - 1; i >= 1; i--)
            {
                int k = rnd.Next(i + 1);
                T tmp = result[k];
                result[k] = result[i];
                result[i] = tmp;
            }
            return result;
        }

        #region Read/Write

        private static TextReader Reader;

        private static TextWriter Writer;

        private static Queue<string> CurrentLineTokens = new Queue<string>();

        private static string[] ReadAndSplitLine()
        {
            return Reader.ReadLine().Split(new[] { ' ', '\t' }, StringSplitOptions.RemoveEmptyEntries);
        }

        public static string ReadToken()
        {
            while (CurrentLineTokens.Count == 0)
                CurrentLineTokens = new Queue<string>(ReadAndSplitLine());
            return CurrentLineTokens.Dequeue();
        }

        public static string ReadLine()
        {
            return Reader.ReadLine();
        }

        public static int ReadInt()
        {
            return int.Parse(ReadToken());
        }

        public static long ReadLong()
        {
            return long.Parse(ReadToken());
        }

        public static double ReadDouble()
        {
            return double.Parse(ReadToken(), CultureInfo.InvariantCulture);
        }

        public static int[] ReadIntArray()
        {
            return ReadAndSplitLine().Select(int.Parse).ToArray();
        }

        public static long[] ReadLongArray()
        {
            return ReadAndSplitLine().Select(long.Parse).ToArray();
        }

        public static double[] ReadDoubleArray()
        {
            return ReadAndSplitLine().Select(s => double.Parse(s, CultureInfo.InvariantCulture)).ToArray();
        }

        public static int[][] ReadIntMatrix(int numberOfRows)
        {
            int[][] matrix = new int[numberOfRows][];
            for (int i = 0; i < numberOfRows; i++)
                matrix[i] = ReadIntArray();
            return matrix;
        }

        public static string[] ReadLines(int quantity)
        {
            string[] lines = new string[quantity];
            for (int i = 0; i < quantity; i++)
                lines[i] = Reader.ReadLine().Trim();
            return lines;
        }

        public static void WriteArray<T>(IEnumerable<T> array)
        {
            Writer.WriteLine(string.Join(" ", array));
        }

        #endregion
    }
}