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 <algorithm>
#include <iostream>
#include <vector>
using namespace std;

const int nmax = 100010;
int deg[nmax],dp[nmax];
bool fix[nmax];
vector < int > graph[nmax];
int N,u;
bool check2(int x,int B, vector<int>& kids)
{
	int l = 0, r = kids.size() - 1;
	if((kids[x]+1) > B) return false;
	while( l < r)
	{
		if(l == x) l++;
		if(r == x) r--;
		if(l >= r  || r < 0) return true;
		if((kids[l] + kids[r]+2 )> B) return false;
		l++,r--;
	}
	return true;
}
bool dfs(int u, int x)
{
	fix[u] = true;
	bool ret = true;
	//cout << "in vertex " << u << " trying " << x << endl;
	vector< int > kids;
	for(int i = 0; i < graph[u].size(); ++i)
	{
		if(fix[graph[u][i]]) continue;
		ret&=dfs(graph[u][i],x);
		kids.push_back(dp[graph[u][i]]);
	}
	if(!ret) return ret;
	bool isfake = false;
	if((kids.size()&1)==0){ isfake = true; kids.push_back(-1);}
	sort(kids.begin(), kids.end());
	int l = 0, r = kids.size() - 1;
	
	//cout << "out to " << u;
	//cout << "\n\t";
	//for(int i = 0 ; i < kids.size(); ++i)
	//	cout << kids[i] << ' ';
	//cout << endl;
	bool isGood = check2(r, x, kids);
	while((r - l) > 1)
	{
		int mid = (r+l) >> 1;
		if(check2(mid, x, kids))
			r = mid;
		else
			l = mid;
	
	}
	if(check2(l, x, kids)) r = l;
	if(!isGood) return false;
	//cout << "\tchose" << kids[r] << endl;
	dp[u] = kids[r]+1;
	return true;
}
bool check(int x)
{
	for(int i = 1;i<=N;++i)dp[i]=N,fix[i]=false;
	return dfs(u,x);
}
int bs()
{
	int l = 1, r = N-1;
	while((r-l) > 1)
	{
		int mid = (l+r) >> 1;
		if(check(mid))
			r = mid;
		else
			l = mid;
		//cout << l << ' ' << r << ' ' << mid << endl;
	}
	if(check(l))
		return l;
	return r;

}
int main()
{
	cin >> N;
	for(int i = 0 ;i < N-1;++i)
	{
		int x,y;
		cin >> x >> y;
		graph[x].push_back(y);
		graph[y].push_back(x);
		deg[x]++;
		deg[y]++;
	}
	int A=0,B=0;
	for(int i = 1;i<=N;++i)
	{
		if(deg[i]&1) A++;
		if(deg[i] == 1) u = i;
	}
	A = (A+1)/2;
	cout << A << ' ' << bs() << endl;
}