Problem: Takahashi has decided to make a Christmas Tree for the Christmas party in AtCoder, Inc.

A Christmas Tree is a tree with N vertices numbered 1 through N and N-1 edges, whose i-th edge (1\leq i\leq N-1) connects Vertex a_i and b_i.

He would like to make one as follows:

* Specify two non-negative integers A and B.
* Prepare A Christmas Paths whose lengths are at most B. Here, a Christmas Path of length X is a graph with X+1 vertices and X edges such that, if we properly number the vertices 1 through X+1, the i-th edge (1\leq i\leq X) will connect Vertex i and i+1.
* Repeat the following operation until he has one connected tree:
* Select two vertices x and y that belong to different connected components. Combine x and y into one vertex. More precisely, for each edge (p,y) incident to the vertex y, add the edge (p,x). Then, delete the vertex y and all the edges incident to y.
* Properly number the vertices in the tree.



Takahashi would like to find the lexicographically smallest pair (A,B) such that he can make a Christmas Tree, that is, find the smallest A, and find the smallest B under the condition that A is minimized.

Solve this problem for him.

Constraints

* 2 \leq N \leq 10^5
* 1 \leq a_i,b_i \leq N
* 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

For the lexicographically smallest (A,B), print A and B with a space in between.

Examples

Input

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


Output

3 2


Input

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


Output

2 5


Input

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


Output

3 4

Solution (in CPP):

Solution:
#include <bits/stdc++.h>

using std::pair;
using std::vector;
using std::string;

typedef long long ll;
typedef pair<int, int> pii;

#define fst first
#define snd second
#define pb(a) push_back(a)
#define mp(a, b) std::make_pair(a, b)
#define debug(...) fprintf(stderr, __VA_ARGS__)

template <typename T> bool chkmax(T& a, T b) { return a < b ? a = b, 1 : 0; }
template <typename T> bool chkmin(T& a, T b) { return a > b ? a = b, 1 : 0; }

const int oo = 0x3f3f3f3f;

template<typename T> T read(T& x) {
    int f = 1; x = 0;
    char ch = getchar();
    for(;!isdigit(ch); ch = getchar()) if(ch == '-') f = -1;
    for(; isdigit(ch); ch = getchar()) x = x * 10 + ch - 48; 
    return x *= f;
}

const int N = 100000;
int st[N + 5], to[(N << 1) + 5], nxt[(N << 1) + 5], e = 1;

inline void addedge(int u, int v) {
    to[++ e] = v; nxt[e] = st[u]; st[u] = e;
    to[++ e] = u; nxt[e] = st[v]; st[v] = e;
}

std::multiset<int> s;
typedef std::multiset<int> :: iterator sit;

int LIM = N + 5;
int f[N + 5], g[N + 5];
void dfs(int u, int fa = 0) {
    f[u] = g[u] = 0;
    for(int i = st[u]; i; i = nxt[i]) {
        int v = to[i];
        if(v != fa) dfs(v, u), f[u] += f[v];
    }

    s.clear();
    for(int i = st[u]; i; i = nxt[i]) {
        int v = to[i];
        if(v != fa) s.insert(g[v] + 1); 
    }
    
    int ret = oo;

    while(s.size()) {
        int x = (*s.rbegin());
        sit it = s.upper_bound(LIM - x);

        ++ f[u];
        if(it == s.begin() || s.size() == 1) {
            if(x < LIM) {
                chkmin(ret, x);
                s.erase(s.find(*s.rbegin()));
            } else {
                s.erase(s.find(*s.rbegin()));
            }
        } else {
            -- it;
            s.erase(s.find(*it));
            s.erase(s.find(*s.rbegin()));
        }
    }
    if(ret != oo && u > 1) g[u] = ret, -- f[u];

}

int x, y, n;
int main() {
#ifdef Wearry
    freopen("data.txt", "r", stdin);
    freopen("ans.txt", "w", stdout);
#endif

    read(n);
    for(int i = 1; i < n; ++i) {
        read(x), read(y);
        addedge(x, y);
    }

    dfs(1);
    int A = f[1];

    int L = 1, R = n;
    while(L < R) {
        LIM = (L + R) >> 1;

        dfs(1);
        if(f[1] > A) L = LIM + 1; else R = LIM;
    }

    printf("%d %d\n", A, R);

    return 0;
}