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 <algorithm>
#include <bitset>
#include <cassert>
#include <chrono>
#include <climits>
#include <cmath>
#include <complex>
#include <cstring>
#include <deque>
#include <functional>
#include <iostream>
#include <list>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <stack>
#include <unordered_map>
#include <unordered_set>
#include <vector>
#include <cstdint>
using namespace std;
typedef long long ll;
#define MP make_pair
#define PB push_back
#define inf 1000000007
#define mod 1000000007
#define rep(i,n) for(int i = 0; i < (int)(n); ++i)
vector<vector<int> > g;
vector<int>c;
int S;
int dfs(int id,int pre,int len){
    multiset<int> p;
    for(auto x:g[id]){
        if(x!=pre){
            int k = dfs(x,id,len);
            if(k==0||k>len){
                return 0;
            }
            p.insert(k);
        }
    }
    if(c[id]==1){
        return 1;
    }
    // cerr << "id: " << id << endl;
    // for(auto x:p){
    //     cerr << x << " ";
    // }
    // cerr << endl;
    if((c[id]%2==0&&id!=S)||(c[id]%2==1&&id==S)){
        multiset<int> q;
        while(p.size()>=2){
            auto x = p.end();
            x--;
            int k = *x;
            auto y = p.upper_bound(len-k);
            if(y==p.begin()){
                q.insert(k);
                p.erase(x);
            }else{
                y--;
                if(x==y){
                    y--;
                }
                p.erase(x);
                p.erase(y);
            }
        }
        for(auto x:p){
            q.insert(x);
        }
        if(q.size()!=1){
            return 0;
        }else{
            int k = (*q.begin());
            if(id!=S){
                return k+1;
            }else{
                return k;
            }
        }
    }else{
        multiset<int> q;
        while(p.size()>=2){
            auto x = p.end();
            x--;
            int k = *x;
            auto y = p.upper_bound(len-k);
            if(y==p.begin()){
                q.insert(k);
                p.erase(x);
            }else{
                y--;
                if(x==y){
                    y--;
                }
                p.erase(x);
                p.erase(y);
            }
        }
        for(auto x:p){
            q.insert(x);
        }
        if(q.size()==0){
            return 1;
        }else if(q.size()==2){
            int k = *q.begin();
            if(id!=S){
                return k+1;
            }else{
                int s = 0;
                for(auto x:q){
                    s += x;
                }
                return s;
            }
        }else{
            return 0;
        }
    }
}
int main(){
    int n;
    cin >> n;
    g.resize(n);
    c.resize(n);
    rep(i,n-1){
        int a,b;
        cin >> a >> b;
        a--;b--;
        g[a].push_back(b);
        g[b].push_back(a);
        c[a]++;
        c[b]++;
    }
    int A = 1;
    int id = 0;
    rep(i,n){
        if(c[i]>2){
            if(c[i]%2==1){
                A += c[i]/2;
            }else{
                A += c[i]/2-1;
            }
        }
        if(c[i]%2==1){
            id = i;

        }
    }
    int S = 0;
    int ok = n-1;
    int ng = 1;
    while(ok-ng>1){
        int mid = (ok+ng)/2;
        int k = dfs(0,-1,mid); 
        //cerr << mid << " " << k << endl;
        if(k==0||k>mid){
            ng = mid;
        }else{
            ok = mid;
        }
    }
    cout << A << " " << ok << endl;
    return 0;
}