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 mod 998244353
#define MOD 1000000007
#define inf 0x3f3f3f3f
#define linf 0x3f3f3f3f3f3f3f3fll
typedef long long ll;
typedef pair<int,int> pii;
typedef unsigned long long ull;
const int maxn=1e5+10;
vector<int> g[maxn];
int n,deg[maxn],pa[maxn];
vector<int> dfn;
void dfs(int x,int fa)
{
	dfn.push_back(x);
	pa[x]=fa;
	for(int i=0;i<g[x].size();i++)
	{
		int to=g[x][i];
		if(to==fa)continue;
		dfs(to,x);
	}
}
int up[maxn];
vector<int> son;
bool check2(int pos,int len)
{
	if(pos>=son.size())return 0;
//	if(len==2)
//	{
//		cout<<"check2 "<<pos<<" "<<len<<" "<<son[pos]<<endl;
//	}
	if(son[pos]+1>len)return 0;
	int l=0,r=son.size()-1;
	for(int i=0;i<son.size()/2;i++)
	{
		if(l==pos)l++;
		if(r==pos)r--;
		if(son[l]+son[r]>len)return 0;
		l++,r--;
	}
	return 1;
}
bool check(int len)
{
	for(int i=0;i<dfn.size();i++)
	{
		int x=dfn[i];
		son.clear();
		for(int j=0;j<g[x].size();j++)
			if(g[x][j]!=pa[x])
			{
		//		if(len==2)
		//		{
		//			cout<<"wtf "<<x<<" "<<g[x][j]<<" "<<up[g[x][j]]+1<<endl;
		//		}
				son.push_back(up[g[x][j]]+1);
			}
		if(!(son.size()&1))son.push_back(0);
		sort(son.begin(),son.end());
		if(len==2)
		{
	//		for(int j=0;j<son.size();j++)cout<<son[j]<<" ";
	//		cout<<"!! "<<x<<endl;
		}
	//	if(!check2(son.size()-1,len))return 0;
		int l=-1,r=son.size();//(l,r]
	//	cout<<"safe here?\n";
		while(l+1<r)
		{
			int mid=(l+r)>>1;
			if(check2(mid,len))r=mid;
			else l=mid;
		}
		if(r==son.size()||son[r]>=len)return 0;
		if(len==2)
		{
	//		cout<<x<<" "<<r<<" "<<son[r]<<endl;
		}
		up[x]=son[r];
	}
	son.clear();
	for(int i=0;i<g[1].size();i++)son.push_back(up[g[1][i]]+1);
	sort(son.begin(),son.end());
	if(son.back()>len)return 0;
	if(son.size()&1)son.pop_back();
	for(int i=0;i<son.size()/2;i++)
		if(son[i]+son[son.size()-1-i]>len)return 0;
	return 1;
}
int main()
{
	scanf("%d",&n);
	for(int i=1;i<n;i++)
	{
		int a,b;
		scanf("%d%d",&a,&b);
		g[a].push_back(b);
		g[b].push_back(a);
		deg[a]++;
		deg[b]++;
	}
	int A=n-1;
	for(int i=1;i<=n;i++)
	{
		A-=deg[i]/2;
	}
	dfs(1,0);
	reverse(dfn.begin(),dfn.end());
	dfn.pop_back();
	int l=0,r=n;//(l,r]
	while(l+1<r)
	{
		int mid=(l+r)>>1;
		if(check(mid))r=mid;
		else l=mid;
	}
	printf("%d %d\n",A,r);
	return 0;
}