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;

const int maxn=1e5+10;
int n,ans,cnt;
vector<int> g[maxn];

void dfs(int pos,int fa,int mid,multiset<int>&root){
	multiset<int> edge;
	for(int i=0;i<g[pos].size();++i)
		if(g[pos][i]!=fa)
			dfs(g[pos][i],pos,mid,edge);
	int rest=0;
	while(!edge.empty()&&*edge.rbegin()==mid){
		multiset<int>::iterator p=edge.end();--p;
		edge.erase(p);
		++cnt;
	}
	while(edge.size()){
		multiset<int>::iterator p=edge.end(),q;--p;
		int maximum=*p;
		edge.erase(p);
		q=edge.upper_bound(mid-maximum);
		if(q==edge.begin()){
			if(rest)
				++cnt,rest=maximum;
			else
				rest=maximum;
		}
		else{
			--q;
			edge.erase(q);
			++cnt;
		}
	}
	root.insert(rest+1);
}

int main(){
	cin>>n;
	for(int i=1,u,v;i<n;++i){
		cin>>u>>v;
		g[u].push_back(v);
		g[v].push_back(u);
	}
	for(int i=2;i<=n;++i)
		ans+=(g[i].size()-1)/2;
	ans+=(g[1].size()+1)/2;
	int l=1,r=n-1;
	while(l<r){
		int mid=l+r>>1;
		cnt=0;
		multiset<int> tmp;
		dfs(1,0,mid,tmp);
		if(*tmp.begin()>1)
			++cnt;
		if(cnt<=ans)
			r=mid;
		else
			l=mid+1;
	}
	cout<<ans<<' '<<l<<endl;
	return 0;
}