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.Collections;
using System.Collections.Specialized;
using System.Linq;
using System.Text;
using System.IO;
using System.Reflection;
using static System.Math;
using System.Numerics;
static class Program{
	const int mod=(int)1e9+7;
	static List<int>[] li;
	static bool[] b;
	static int[] f,h;
	static void Main(){
		Sc sc=new Sc();
		var s=sc.Ia;
		int n=s[0],m=n-1;
		li=new List<int>[n+1];
		b=new bool[n+1];
		f=new int[n+1];
		for(int i=1;i<=n;i++){li[i]=new List<int>();}
		for(int i=0;i<m;i++){
			var e=sc.Ia;
			li[e[0]].Add(e[1]);
			li[e[1]].Add(e[0]);
		}
		h=new int[n+1];
		var c1=Fc1(1,1);
		h=new int[n+1];
		var c2=Fc2(c1,1);
		long ans=0;
		if(h[c2.Item1]%2==0){
			f=new int[(h[c2.Item1]>>1)+1];
			h=new int[n+1];
			h[c2.Item3]=1;
			Fu(c2.Item2,1);
			Fu(c2.Item3,1);
			ans=2;
			for(int i = 1;f[i]!=0;i++) {ans*=f[i];}
			Console.WriteLine("{0} {1}",f.Length-1,ans);
		}
		else{
			f=new int[(h[c2.Item1]>>1)+2];
			int p=f.Length-1;
			h=new int[n+1];
			Fu(c2.Item2,1);
			ans=li[c2.Item2].Count;
			for(int i = 2;f[i]!=0;i++) {ans*=f[i];}
			for(int i = 0;i<li[c2.Item2].Count;i++) {
				f=new int[p+1];
				h=new int[n+1];
				h[c2.Item2]=1;
				Fu(li[c2.Item2][i],1);
				Fu(c2.Item2,1);
				long z=2;
				for(int j = 1;f[j]!=0;j++) {z*=f[j];}
				ans=Min(ans,z);
			}
			Console.WriteLine("{0} {1}",p,ans);
		}
	}
	static void Fu(int a,int g){
		h[a]=g;
		f[g]=Max(f[g],li[a].Count-1);
		for(int i=0;i<li[a].Count;i++){
			if(h[li[a][i]]==0){Fu(li[a][i],g+1);}
		}
	}
	static int Fc1(int a,int g){
		h[a]=g;
		int c=a;
		for(int i=0;i<li[a].Count;i++){
			if(h[li[a][i]]==0){
				var d=Fc1(li[a][i],g+1);
				if(h[c]<h[d]){c=d;}
			}
		}
		return c;
	}
	static Tuple<int,int,int> Fc2(int a,int g){
		h[a]=g;
		int c=a,e=-1,f=-1;
		for(int i=0;i<li[a].Count;i++){
			if(h[li[a][i]]==0){
				var d=Fc2(li[a][i],g+1);
				if(h[c]<h[d.Item1]){c=d.Item1;e=d.Item2;f=d.Item3;}
			}
		}
		if((h[c]&1)==0){
			if((h[c])>>1==g){e=a;}
			else if(((h[c])>>1)+1==g){f=a;}
		}
		else if((h[c]+1)>>1==g){e=a;}
		return Tuple.Create(c,e,f);
	}
}
public class Sc{
	public int I{get{return int.Parse(Console.ReadLine());}}
	public long L{get{return long.Parse(Console.ReadLine());}}
	public double D{get{return double.Parse(Console.ReadLine());}}
	public string S{get{return Console.ReadLine();}}
	public int[] Ia{get{return Array.ConvertAll(Console.ReadLine().Split(),int.Parse);}}
	public long[] La{get{return Array.ConvertAll(Console.ReadLine().Split(),long.Parse);}}
	public double[] Da{get{return Array.ConvertAll(Console.ReadLine().Split(),double.Parse);}}
	public string[] Sa{get{return Console.ReadLine().Split();}}
	public object[] Oa{get{return Console.ReadLine().Split();}}
	public int[] Ia2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),int.Parse);}}
	public int[] Ia3(int a){return Array.ConvertAll((a.ToString()+" "+Console.ReadLine()).Split(),int.Parse);}
	public int[] Ia3(bool a,int b,bool c,int d){return Array.ConvertAll(((a?b.ToString()+" ":"")+Console.ReadLine()+(c?" "+d.ToString():"")).Split(),int.Parse);}
	public long[] La2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),long.Parse);}}
	public long[] La3(int a){return Array.ConvertAll((a.ToString()+" "+Console.ReadLine()).Split(),long.Parse);}
	public long[] La3(bool a,int b,bool c,int d){return Array.ConvertAll(((a?b.ToString()+" ":"")+Console.ReadLine()+(c?" "+d.ToString():"")).Split(),long.Parse);}
	public T[] Arr<T>(int n,Func<T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f();}return a;}
	public T[] Arr<T>(int n,Func<int,T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(i);}return a;}
	public T[] Arr<T>(int n,Func<string[],T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(Console.ReadLine().Split());}return a;}
	public T[] Arr<T>(int n,Func<int,string[],T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(i,Console.ReadLine().Split());}return a;}
}