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 Java8 (OpenJDK 1.8.0) (judge verdict: Runtime Error)

import java.io.*;
import java.util.*;
import java.math.*;
// import java.awt.Point;
 
public class Main {
    InputStream is;
    PrintWriter out;
    String INPUT = "";
 
    long mod = 1_000_000_007;
    long inf = Long.MAX_VALUE;

    void solve(){
        int n = ni();
        ArrayList<Integer>[] edge = new ArrayList[n];
        int[][] e = new int[n-1][2];
        for(int i = 0; i < n; i++) edge[i] = new ArrayList<Integer>();
        for(int i = 0; i < n-1; i++){
            int from = ni()-1;
            int to = ni()-1;
            edge[from].add(to);
            edge[to].add(from);
            e[i][0] = from;
            e[i][1] = to;
        }
        int col = n;
        long[] le = new long[n];
        Arrays.fill(le,inf);
        for(int i = 0; i < n; i++){
            boolean[] seen = new boolean[n];
            Deque<int[]> q = new ArrayDeque<>();
            q.add(new int[]{i, 0});
            int[] leaf = new int[n+1];
            int cmax = 0;
            while(!q.isEmpty()){
                int[] res = q.pop();
                cmax = Math.max(cmax, res[1]+1);
                seen[res[0]] = true;
                int p = 0;
                for(int j : edge[res[0]]){
                    if(!seen[j]){
                        q.add(new int[]{j, res[1]+1});
                        p++;
                    }
                }
                leaf[res[1]] = Math.max(leaf[res[1]], p);
            }
            long lt = 1;
            for(int j = 0; j < n; j++){
                if(leaf[j]==0) break;
                lt *= leaf[j];
            }
            le[cmax] = Math.min(le[cmax], lt);
            col = Math.min(cmax, col);
        }
        for(int i = 0; i < n-1; i++){
            boolean[] seen = new boolean[n];
            Deque<int[]> q = new ArrayDeque<>();
            q.add(new int[]{e[i][0], 0});
            q.add(new int[]{e[i][1], 0});
            seen[e[i][0]] = true;
            seen[e[i][1]] = true;
            int[] leaf = new int[n+1];
            int cmax = 0;
            while(!q.isEmpty()){
                int[] res = q.pop();
                cmax = Math.max(cmax, res[1]+1);
                seen[res[0]] = true;
                int p = 0;
                for(int j : edge[res[0]]){
                    if(!seen[j]){
                        q.add(new int[]{j, res[1]+1});
                        p++;
                    }
                }
                leaf[res[1]] = Math.max(leaf[res[1]], p);
            }
            long lt = 2;
            for(int j = 0; j < n; j++){
                if(leaf[j]==0) break;
                lt *= leaf[j];
            }
            le[cmax] = Math.min(le[cmax], lt);
            col = Math.min(cmax, col);
        }
        out.println(col+" "+le[col]);
    }


    void run() throws Exception
    {
        is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
        out = new PrintWriter(System.out);
        long s = System.currentTimeMillis();
        solve();
        out.flush();
        if(!INPUT.isEmpty())tr(System.currentTimeMillis()-s+"ms");
    }
    
    public static void main(String[] args) throws Exception { new Main().run(); }
    
    private byte[] inbuf = new byte[1024];
    private int lenbuf = 0, ptrbuf = 0;
    
    private int readByte()
    {
        if(lenbuf == -1)throw new InputMismatchException();
        if(ptrbuf >= lenbuf){
            ptrbuf = 0;
            try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
            if(lenbuf <= 0)return -1;
        }
        return inbuf[ptrbuf++];
    }
    
    private boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
    private int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
    
    private double nd() { return Double.parseDouble(ns()); }
    private char nc() { return (char)skip(); }
    
    private String ns()
    {
        int b = skip();
        StringBuilder sb = new StringBuilder();
        while(!(isSpaceChar(b) && b != ' ')){
            sb.appendCodePoint(b);
            b = readByte();
        }
        return sb.toString();
    }
    
    private char[] ns(int n)
    {
        char[] buf = new char[n];
        int b = skip(), p = 0;
        while(p < n && !(isSpaceChar(b))){
            buf[p++] = (char)b;
            b = readByte();
        }
        return n == p ? buf : Arrays.copyOf(buf, p);
    }
    
    private char[][] nm(int n, int m)
    {
        char[][] map = new char[n][];
        for(int i = 0;i < n;i++)map[i] = ns(m);
        return map;
    }
    
    private int[] na(int n)
    {
        int[] a = new int[n];
        for(int i = 0;i < n;i++)a[i] = ni();
        return a;
    }
    
    private int ni()
    {
        int num = 0, b;
        boolean minus = false;
        while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
        if(b == '-'){
            minus = true;
            b = readByte();
        }
        
        while(true){
            if(b >= '0' && b <= '9'){
                num = num * 10 + (b - '0');
            }else{
                return minus ? -num : num;
            }
            b = readByte();
        }
    }
    
    private long nl()
    {
        long num = 0;
        int b;
        boolean minus = false;
        while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
        if(b == '-'){
            minus = true;
            b = readByte();
        }
        
        while(true){
            if(b >= '0' && b <= '9'){
                num = num * 10 + (b - '0');
            }else{
                return minus ? -num : num;
            }
            b = readByte();
        }
    }
    
    private static void tr(Object... o) { System.out.println(Arrays.deepToString(o)); }
 
}