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<cstdio>
#include<cstring>
#include<algorithm>
#include<cmath>
#include<set>
#include<vector>
#define SF scanf
#define PF printf
#define MAXN 100010
using namespace std;
int dp[MAXN];
int l,r,n,sum,ans;
multiset<int> s;
vector<int> a[MAXN];
bool dfs(int x,int fa,int maxl){
    for(int i=0;i<a[x].size();i++)
        if(a[x][i]!=fa)
            if(dfs(a[x][i],x,maxl)==0)
                return 0;
    s.clear();
    for(int i=0;i<a[x].size();i++)
        if(a[x][i]!=fa){
            if(dp[a[x][i]]+1>maxl){
                return 0;
            }
            s.insert(dp[a[x][i]]+1);
        }
    int flag=0;
    if(s.size()%2==1)
        flag=1;
    if(x==1)
        flag--;
    /*if(x==83)
        for(int i=0;i<a[x].size();i++)
            if(a[x][i]!=fa)
                PF("{%d} ",dp[a[x][i]]+1);*/
    while(s.size()>1){
        int t=*s.rbegin();
        s.erase(s.find(t));
        multiset<int>::iterator it=s.lower_bound(maxl-t+1);
        if(it==s.begin()){
            if(flag==-1){
                return 0;
            }
            if(flag==1){
                dp[x]=t;
                flag=-1;
            }
            else{
                if(x!=1)
                    flag=1;
                else
                    flag=-1;
            }
        }
        else{
            it--;
            s.erase(it);
        }
    }
    if(s.size()==1){
        dp[x]=*s.begin();
        if(dp[x]>maxl){
            return 0;
        }
        s.erase(dp[x]);
    }
    return 1;
}
bool check(int maxl){
    memset(dp,0,sizeof dp);
    s.clear();
    if(dfs(1,0,maxl))
        return 1;
    return 0;
}
int u,v;
int main(){
    //freopen("data.in","r",stdin);
    //freopen("data.out","w",stdout);
    SF("%d",&n);
    for(int i=1;i<n;i++){
        SF("%d%d",&u,&v);
        a[u].push_back(v);
        a[v].push_back(u);
    }
    for(int i=1;i<=n;i++)
        if(a[i].size()%2==1)
            sum++;
    sum/=2;
    PF("%d ",sum);
    //PF("[%d]",check(6));
    l=1,r=n;
    while(l<=r){
        int mid=(l+r)>>1;
        if(check(mid)){
            ans=mid;
            r=mid-1;
        }
        else{
            l=mid+1;
        }
    }
    PF("%d",ans);
}