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++14 (Clang 3.8.0) (judge verdict: Wrong Answer)

/*

This Submission is to determine how many 120/240 min const. delivery point there are.

//info
120 req. steps <= 5
*/
#define  _CRT_SECURE_NO_WARNINGS
#include <stdio.h>
#include <algorithm>
#include <utility>
#include <functional>
#include <cstring>
#include <queue>
#include <stack>
#include <math.h>
#include <iterator>
#include <vector>
#include <string>
#include <set>
#include <math.h>
#include <iostream>
#include <random>
#include<map>
#include <iomanip>
#include <time.h>
#include <stdlib.h>
#include <list>
#include <typeinfo>
#include <list>
#include <set>
#include <cassert>
#include<fstream>
#include <unordered_map>
#include <cstdlib>
#include <complex>
#include <cctype>
#include <bitset>
using namespace std;
typedef string::const_iterator State;
#define Ma_PI 3.141592653589793
#define eps 1e-5
#define LONG_INF 1e18
#define GOLD 1.61803398874989484820458
#define MAX_MOD 1000000007
#define MOD 998244353
#define seg_size 262144
#define REP(a,b) for(long long a = 0;a < b;++a)

unsigned long xor128() {
	static unsigned long x = time(NULL), y = 362436069, z = 521288629, w = 88675123;
	unsigned long t = (x ^ (x << 11));
	x = y; y = z; z = w;
	return (w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)));
}
double dot(complex<double> a, complex<double> b) {
	return a.real() * b.real() + a.imag() * b.imag();
}
double gyaku_dot(complex<double> a, complex<double> b) {
	return a.real() * b.imag() - a.imag() * b.real();
}
double leng(complex<double> a) {
	return sqrt(a.real() * a.real() + a.imag() * a.imag());
}
double angles(complex<double> a, complex<double> b) {
	double cosine = dot(a, b) / (leng(a) * leng(b));
	double sine = gyaku_dot(a, b) / (leng(a) * leng(b));
	double kaku = acos(min((double)1.0, max((double)-1.0, cosine)));
	if (sine <= 0) {
		kaku = 2 * Ma_PI - kaku;
	}
	return kaku;
}
vector<int> convex_hull(vector<complex<double>> a) {
	vector<int> ans;
	double now_minnest = a[0].real();
	int now_itr = 0;
	REP(i, a.size()) {
		if (now_minnest > a[i].real()) {
			now_minnest = a[i].real();
			now_itr = i;
		}
	}
	ans.push_back(now_itr);
	complex<double> ba(0, 1);
	while (true) {
		int now_go = 0;
		double now_min = 0;
		double now_length = 0;
		int starter = ans[ans.size() - 1];
		for (int i = 0; i < a.size(); ++i) {
			if (i != starter) {
				double goa = angles(ba, a[i] - a[starter]);
				if (goa - now_min >= eps || (abs(goa - now_min) <= eps && (abs(a[i] - a[starter]) - now_length) >= eps)) {
					now_min = goa;
					now_go = i;
					now_length = abs(a[i] - a[starter]);
				}
			}
		}
		if (now_go == ans[0]) break;
		ans.push_back(now_go);
		ba = complex<double>(a[now_go] - a[starter]);
	}
	return ans;
}
#define int long long
vector<int> vertexs[200];
int distances[2000];
void dfs(int now, int back) {
	REP(q, vertexs[now].size()) {
		int go = vertexs[now][q];
		if (go == back) continue;
		distances[go] = distances[now] + 1;
		dfs(go, now);
	}
}
long long visited[2000] = {};
long long bobo[2000];
long long solve(long long a, long long b) {
	for (int i = 0; i < 2000; ++i) {
		visited[i] = 0;
		distances[i] = 0;
		bobo[i] = 0;
	}
	dfs(a, b);
	dfs(b, a);
	for (int i = 0;vertexs[i].size() != 0; ++i) {
		int hoge = 0;
		if (a != b||distances[i] !=0) {
			hoge = 1;
		}
		bobo[distances[i]] = max(bobo[distances[i]], (long long)vertexs[i].size() - hoge);
	}
	long long ans = 1;
	for (int i = 0; bobo[i] != 0; ++i) {
		ans *= bobo[i];
	}
	if (a != b) {
		ans *= 2LL;
	}
	return ans;
}
#undef int
int main() {
#define int long long
	int n;
	cin >> n;
	REP(i, n - 1) {
		int a, b;
		cin >> a >> b;
		a--;
		b--;
		vertexs[a].push_back(b);
		vertexs[b].push_back(a);
	}
	dfs(0, -1);
	int now_max = 0;
	int now_itr = 0;
	REP(i, n) {
		if (distances[i] > now_max) {
			now_max = distances[i];
			now_itr = i;
		}
		distances[i] = 0;
	}
	dfs(now_itr, -1);
	REP(i, n) {
		now_max = max(now_max, distances[i]);
	}
	cout << (now_max + 2) / 2;
	vector<pair<int, int>> kouho;
	if (now_max % 2 == 1) {
		for (int q = 0; q < n; ++q) {
			if (distances[q] != now_max / 2) continue;
			for (int j = 0; j < vertexs[q].size(); ++j) {
				int go = vertexs[q][j];
				if (distances[go] == now_max / 2 + 1) {
					kouho.push_back(make_pair(q, go));
				}
			}
		}
	}
	else {
		for (int q = 0; q < n; ++q) {
			if (distances[q] != now_max / 2) continue;
			kouho.push_back(make_pair(q, q));
			for (int j = 0; j < vertexs[q].size(); ++j) {
				int go = vertexs[q][j];
				kouho.push_back(make_pair(q, go));
			}
		}
	}
	long long ans = 2e18;
	REP(i, kouho.size()) {
		ans = min(ans, solve(kouho[i].first, kouho[i].second));
	}
	cout << " " << ans << endl;
}