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>

#define debug(x) cerr << #x << ": " << x << '\n'
#define debugArray(x,n) for(long long hoge = 0; (hoge) < (n); ++ (hoge)) cerr << #x << "[" << hoge << "]: " << x[hoge] << '\n'

using namespace std;

typedef long long ll;
typedef unsigned long long ull;
typedef pair<ll,ll> Pll;
typedef vector<ll> vll;
const ll INF = LLONG_MAX/10;
const ll MOD = 1e9+7;

ll N;
vector<vll> G;
ll degree[100010];

ll dp[100010];
bool dfs(ll v,ll p,const ll &x){
  bool ret=true;
  vll V;
  for(ll u:G[v])if(p!=u){
    ret &= dfs(u,v,x);
    V.push_back(dp[u]+1);
  }
  if(!V.size())return ret;
  if(!ret)return false;
  sort(V.begin(),V.end());
  if(V.size()&1){
    ll low=-1,high=V.size();
    while(high-low>1){
      ll mid = (low+high)/2;
      bool isok=V[mid]<=x;
      ll l=0,r=V.size()-1;
      for(ll i=0;i<V.size()/2;i++){
        if(l==mid){
          l++;
        }
        if(r==mid){
          r--;
        }
        isok &= V[l]+V[r]<=x;
        l++;
        r--;
      }
      if(isok){
        high = mid;
      }else{
        low = mid;
      }
    }
    if(high==V.size()){
      return false;
    }
    dp[v] = V[high];
  }else{
    bool isok=true;
    ll l=0,r=V.size()-1;
    for(ll i=0;i<V.size()/2;i++){
      isok &= V[l]+V[r]<=x;
      l++;
      r--;
    }
    if(!isok){
      //cerr<<"!!"<<endl;
      ret &= V.back()<=x;
      V.pop_back();
      ll low=-1,high=V.size();
      while(high-low>1){
        ll mid = (low+high)/2;
        isok=V[mid]<=x;
        l=0,r=V.size()-1;
        for(ll i=0;i<V.size()/2;i++){
          if(l==mid){
            l++;
          }
          if(r==mid){
            r--;
          }
          isok &= V[l]+V[r]<=x;
          l++;
          r--;
        }
        if(isok){
          high = mid;
        }else{
          low = mid;
        }
      }
      if(high==V.size()){
        return false;
      }
      dp[v] = V[high];
    }
  }
  ret &= dp[v]<=x;
  return ret;
}

int main(){
  cin.tie(0);
  ios::sync_with_stdio(false);
  cin>>N;
  G.resize(N);
  for(ll i=0;i<N-1;i++){
    ll a,b;cin>>a>>b;a--;b--;
    G[a].push_back(b);
    G[b].push_back(a);
    degree[a]++;
    degree[b]++;
  }
  int root=0;
  ll A=0;
  for(ll i=0;i<N;i++){
    if(degree[i]&1){
      A++;
      root=i;
    }
  }
  A /= 2;
  ll low=-1,high=N;
  while(high-low>1){
    ll mid=(high+low)/2;
    //debug(mid);
    fill(dp,dp+N,0);
    if(dfs(root,-1,mid)){
      high = mid;
    }else{
      low = mid;
    }
    //debugArray(dp,N);
  }
  cout<<A<<" "<<high<<endl;
  return 0;
}