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 namespace std;
#define GET_MACRO(_1, _2, _3, NAME, ...) NAME
#define _repl(i,a,b) for(int i=(int)(a);i<(int)(b);i++)
#define _rep(i,n) _repl(i,0,n)
#define rep(...) GET_MACRO(__VA_ARGS__, _repl, _rep)(__VA_ARGS__)
#define mp(a,b) make_pair((a),(b))
#define pb(a) push_back((a))
#define all(x) (x).begin(),(x).end()
#define uniq(x) sort(all(x)),(x).erase(unique(all(x)),end(x))
#define fi first
#define se second
#define dbg(...) _dbg(#__VA_ARGS__, __VA_ARGS__)
void _dbg(string){cout<<endl;}
template<class H,class... T> void _dbg(string s,H h,T... t){int l=s.find(',');cout<<s.substr(0,l)<<" = "<<h<<", ";_dbg(s.substr(l+1),t...);}
template<class T,class U> ostream& operator<<(ostream &o, const pair<T,U> &p){o<<"("<<p.fi<<","<<p.se<<")";return o;}
template<class T> ostream& operator<<(ostream &o, const vector<T> &v){o<<"[";for(T t:v){o<<t<<",";}o<<"]";return o;}

#define INF 1120000000

vector<int> g[100005];

int dfs(int m, int d, int from){
  vector<int> ch;
  for(auto to : g[d]) if(to!=from){
    int r = dfs(m, to ,d);
    if(r > m) return INF;
    ch.pb(r);
  }
  // dbg(m,d,ch);
  int sz = ch.size();
  if(sz==0) return 1;
  if(sz==1) return ch[0]+1;

  sort(all(ch));
  auto check = [&](int mid){
    int l = 0, r = sz-1;
    while(l<r){
      if(l==mid) l++;
      if(r==mid) r--;
      if(l>=r) break;
      if(ch[l]+ch[r] > m) return false;
      l++;
      r--;
    }
    return true;
  };
  if(sz%2==0){
    if(check(-1)) return 1;
    else sz--;
  }

  if(check(0)) return ch[0]+1;
  if(!check(sz-1)) return INF;
  int l = 0, r = sz-1;
  while(r-l>1){
    int mid = (l+r)/2;
    if(check(mid)) r = mid;
    else l = mid;
  }
  return ch[r]+1;
}

int main(){
  int n;
  cin>>n;
  rep(i,n-1){
    int a,b;
    cin>>a>>b;
    a--;b--;
    g[a].pb(b);
    g[b].pb(a);
  }

  int a = 1;
  rep(i,n) a += (g[i].size()-1)/2;

  int d = 0;
  while(g[d].size()!=1) d++;

  int l = 0, r = n;
  while(r-l>1){
    int m = (r+l)/2;
    if(dfs(m, d, -1) <= m+1) r = m;
    else l = m;
  }

  cout << a << " " << r << endl;

  return 0;
}