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 Rust (1.15.1) (judge verdict: Accepted)

use self::mod_int::ModInt;
use std::cmp::{max, min};
use std::collections::{BTreeSet, BinaryHeap, VecDeque};

const MOD: usize = 1e9 as usize + 7;

fn main() {
    let (r, w) = (std::io::stdin(), std::io::stdout());
    let mut sc = IO::new(r.lock(), w.lock());

    let n: usize = sc.read();
    let mut graph = vec![vec![]; n];
    for _ in 1..n {
        let a = sc.read::<usize>() - 1;
        let b = sc.read::<usize>() - 1;
        graph[a].push(b);
        graph[b].push(a);
    }

    let mut ans = vec![];
    for i in 0..n {
        ans.push(solve(&graph, i, i));
        for &next in graph[i].iter() {
            ans.push(solve(&graph, i, next));
        }
    }
    ans.sort();
    let (color, leaves) = ans[0];
    println!("{} {}", color, leaves);
}

fn solve(graph: &Vec<Vec<usize>>, c1: usize, c2: usize) -> (usize, usize) {
    let n = graph.len();
    let dist = bfs(&graph, c1, c2);
    let mut max_dist = *dist.iter().max().unwrap();
    let mut max_children = vec![0; max_dist + 1];
    for i in 0..n {
        let dist = dist[i];
        max_children[dist] = max(max_children[dist], graph[i].len() - 1);
    }
    if c1 == c2 {
        max_children[0] = graph[c1].len();
    }
    let mut cur = if c1 == c2 { 1 } else { 2 };
    for i in 0..max_dist {
        cur *= max_children[i];
    }
    (max_dist + 1, cur)
}

fn get_centers(graph: &Vec<Vec<usize>>) -> (usize, usize) {
    let n = graph.len();
    let dist0 = bfs(graph, 0, 0);

    let root1 = dist0
        .into_iter()
        .enumerate()
        .map(|(i, d)| (d, i))
        .max()
        .unwrap()
        .1;
    let dist1 = bfs(graph, root1, root1);

    let root2 = dist1
        .iter()
        .enumerate()
        .map(|(i, &d)| (d, i))
        .max()
        .unwrap()
        .1;
    let diameter = dist1[root2];
    let dist2 = bfs(graph, root2, root2);
    let mut c1 = 0;
    let mut c2 = 0;
    for i in 0..n {
        if dist1[i] + dist2[i] == diameter {
            if dist1[i] * 2 == diameter {
                c1 = i;
                c2 = i;
            } else if dist1[i] == diameter / 2 {
                c1 = i;
            } else if dist2[i] == diameter / 2 {
                c2 = i;
            }
        }
    }
    (c1, c2)
}

fn bfs(graph: &Vec<Vec<usize>>, root1: usize, root2: usize) -> Vec<usize> {
    let n = graph.len();
    let mut dist = vec![n; n];
    dist[root1] = 0;
    dist[root2] = 0;
    let mut q = VecDeque::new();
    q.push_back(root1);
    q.push_back(root2);
    while let Some(v) = q.pop_front() {
        for &next in graph[v].iter() {
            if dist[next] > dist[v] + 1 {
                dist[next] = dist[v] + 1;
                q.push_back(next);
            }
        }
    }
    dist
}

fn update_min<T: Ord>(a: &mut T, b: T) {
    if *a > b {
        *a = b;
    }
}

pub mod fenwick_tree {
    /// `FenwickTree` is a data structure that can efficiently update elements
    /// and calculate prefix sums in a table of numbers.
    /// [https://en.wikipedia.org/wiki/Fenwick_tree](https://en.wikipedia.org/wiki/Fenwick_tree)
    pub struct FenwickTree<T> {
        n: usize,
        data: Vec<T>,
        init: T,
    }

    impl<T: Copy + ::std::ops::AddAssign + ::std::ops::Sub<Output = T>> FenwickTree<T> {
        /// Constructs a new `FenwickTree`. The size of `FenwickTree` should be specified by `size`.
        pub fn new(size: usize, init: T) -> FenwickTree<T> {
            FenwickTree {
                n: size + 1,
                data: vec![init; size + 1],
                init: init,
            }
        }

        pub fn add(&mut self, k: usize, value: T) {
            let mut x = k;
            while x < self.n {
                self.data[x] += value;
                x |= x + 1;
            }
        }

        /// Returns a sum of range `[l, r)`
        pub fn sum(&self, l: usize, r: usize) -> T {
            self.sum_one(r) - self.sum_one(l)
        }

        /// Returns a sum of range `[0, k)`
        pub fn sum_one(&self, k: usize) -> T {
            assert!(k < self.n, "k={} n={}", k, self.n);

            let mut result = self.init;
            let mut x = k as i32 - 1;
            while x >= 0 {
                result += self.data[x as usize];
                x = (x & (x + 1)) - 1;
            }

            result
        }
    }
}

pub struct CumulativeSum<T> {
    ny: usize,
    nx: usize,
    sum: Vec<Vec<T>>,
}

impl<T> CumulativeSum<T>
where
    T: Copy + std::ops::Add<Output = T> + std::ops::Sub<Output = T>,
{
    pub fn new(a: &Vec<Vec<T>>, init: T) -> CumulativeSum<T> {
        assert!(a.len() > 0);
        let ny = a.len();
        let nx = a[0].len();
        let mut sum = vec![vec![init; nx + 1]; ny + 1];
        for i in 0..ny {
            for j in 0..nx {
                sum[i + 1][j + 1] = a[i][j] + sum[i][j + 1] + sum[i + 1][j] - sum[i][j];
            }
        }
        CumulativeSum {
            ny: ny,
            nx: nx,
            sum: sum,
        }
    }

    pub fn get_sum(&self, y1: usize, x1: usize, y2: usize, x2: usize) -> T {
        assert!(y1 <= y2 && x1 <= x2);
        assert!(y2 <= self.ny - 1);
        assert!(x2 <= self.nx - 1);
        return self.sum[y2 + 1][x2 + 1] + self.sum[y1][x1]
            - self.sum[y1][x2 + 1]
            - self.sum[y2 + 1][x1];
    }
}

pub struct Combination {
    fact: Vec<usize>,
    inv_fact: Vec<usize>,
    modulo: usize,
}

impl Combination {
    pub fn new(max: usize, modulo: usize) -> Combination {
        let mut inv = vec![0; max + 1];
        let mut fact = vec![0; max + 1];
        let mut inv_fact = vec![0; max + 1];
        inv[1] = 1;
        for i in 2..(max + 1) {
            inv[i] = inv[modulo % i] * (modulo - modulo / i) % modulo;
        }
        fact[0] = 1;
        inv_fact[0] = 1;
        for i in 0..max {
            fact[i + 1] = fact[i] * (i + 1) % modulo;
        }
        for i in 0..max {
            inv_fact[i + 1] = inv_fact[i] * inv[i + 1] % modulo;
        }
        Combination {
            fact: fact,
            inv_fact: inv_fact,
            modulo: modulo,
        }
    }

    pub fn get(&self, x: usize, y: usize) -> ModInt<usize> {
        assert!(x >= y);
        ModInt(self.fact[x] * self.inv_fact[y] % self.modulo * self.inv_fact[x - y] % self.modulo)
    }
}

pub mod mod_int {
    use super::MOD;
    use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Sub, SubAssign};

    type Num = usize;

    #[derive(Clone, Copy)]
    pub struct ModInt<T: Copy + Clone>(pub T);

    impl Add<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self + rhs.0
        }
    }

    impl Add<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, mut rhs: Num) -> ModInt<Num> {
            if rhs >= MOD {
                rhs %= MOD;
            }
            let mut t = rhs + self.0;
            if t >= MOD {
                t = t - MOD;
            }
            ModInt(t)
        }
    }

    impl Sub<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, rhs: Num) -> ModInt<Num> {
            let rhs = if rhs >= MOD { rhs % MOD } else { rhs };
            let value = if self.0 < rhs { self.0 + MOD } else { self.0 };
            ModInt(value - rhs)
        }
    }

    impl Sub<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self - rhs.0
        }
    }

    impl AddAssign<Num> for ModInt<Num> {
        fn add_assign(&mut self, other: Num) {
            *self = *self + other;
        }
    }
    impl AddAssign<ModInt<Num>> for ModInt<Num> {
        fn add_assign(&mut self, other: ModInt<Num>) {
            *self = *self + other;
        }
    }

    impl SubAssign<Num> for ModInt<Num> {
        fn sub_assign(&mut self, other: Num) {
            *self = *self - other;
        }
    }

    impl SubAssign<ModInt<Num>> for ModInt<Num> {
        fn sub_assign(&mut self, other: ModInt<Num>) {
            *self = *self - other;
        }
    }

    impl Div<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn div(self, mut rhs: Num) -> ModInt<Num> {
            if rhs >= MOD {
                rhs %= MOD;
            }
            self * ModInt(rhs).pow(MOD - 2)
        }
    }

    impl Div<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn div(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self / rhs.0
        }
    }

    impl DivAssign<Num> for ModInt<Num> {
        fn div_assign(&mut self, rhs: Num) {
            *self = *self / rhs
        }
    }
    impl DivAssign<ModInt<Num>> for ModInt<Num> {
        fn div_assign(&mut self, rhs: ModInt<Num>) {
            *self = *self / rhs
        }
    }

    impl Mul<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;

        fn mul(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self * rhs.0
        }
    }
    impl Mul<Num> for ModInt<Num> {
        type Output = ModInt<Num>;

        fn mul(self, mut rhs: Num) -> ModInt<Num> {
            if rhs >= MOD {
                rhs %= MOD;
            }
            let t = (self.0 * rhs) % MOD;
            ModInt(t)
        }
    }

    impl MulAssign<Num> for ModInt<Num> {
        fn mul_assign(&mut self, rhs: Num) {
            *self = *self * rhs;
        }
    }

    impl MulAssign<ModInt<Num>> for ModInt<Num> {
        fn mul_assign(&mut self, rhs: ModInt<Num>) {
            *self = *self * rhs;
        }
    }

    impl ModInt<Num> {
        pub fn pow(self, e: usize) -> ModInt<Num> {
            let mut result = ModInt(1);
            let mut cur = self;
            let mut e = e;
            while e > 0 {
                if e & 1 == 1 {
                    result *= cur;
                }
                e >>= 1;
                cur *= cur;
            }
            result
        }
    }
}

pub struct IO<R, W: std::io::Write>(R, std::io::BufWriter<W>);

impl<R: std::io::Read, W: std::io::Write> IO<R, W> {
    pub fn new(r: R, w: W) -> Self {
        IO(r, std::io::BufWriter::new(w))
    }
    pub fn write<S: ToString>(&mut self, s: S) {
        use std::io::Write;
        self.1.write(s.to_string().as_bytes()).unwrap();
    }
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .0
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r' || b == b'\t')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r' && b != b'\t')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}