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 <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <memory.h>
#include <math.h>
#include <assert.h>
#include <queue>
#include <map>
#include <set>
#include <string>
#include <algorithm>
#include <iostream>
#include <functional>
#include <unordered_map>
#include <unordered_set>
#include <list>
#include <bitset>

using namespace std;
typedef pair<int, int> pii;
typedef long long ll;
#define Fi first
#define Se second
#define pb(x) push_back(x)
#define szz(x) ((int)(x).size())
#define rep(i, n) for(int i=0;i<n;i++)
#define all(x) (x).begin(), (x).end()
typedef tuple<int, int, int> t3;
typedef pair<ll, ll> pll;
typedef long double ldouble;
typedef pair<double, double> pdd;

int n;
vector <int> E[200020];
int d[200020];

int good(vector <int> h, int mxl) {
	rep(i, szz(h)/2) if(h[i] + h[szz(h)-1-i] > mxl) return 0;
	return 1;
}

int dfs(int x, int fa, int mxl) {
	if(szz(E[x]) - (fa != -1) == 0) { d[x] = 0; return 1; }
	vector <int> L;
	for(int e : E[x]) if(e != fa) {
		if(!dfs(e, x, mxl)) return 0;
		L.pb(d[e]);
	}
	sort(all(L));
	if(szz(L) % 2 == 0) {
		if(good(L, mxl - 2)) { d[x] = 0; return 1; }
		if(fa == -1) { return 0; }
		int low = 0, high = szz(L) - 1, res = -1;
		while(low <= high) {
			int mid = (low + high) >> 1;
			vector <int> nh;
			rep(i, szz(L)) if(i != mid) nh.pb(L[i]);
			int ok = 1;
			if(nh.back() + 1 > mxl) ok = 0;
			nh.pop_back();
			if(good(nh, mxl - 2) == 0) ok = 0;
			if(ok) res = mid, high = mid - 1;
			else low = mid + 1;
		}
		if(res == -1 || L[res] + 1 > mxl) return 0;
		d[x] = L[res] + 1;
		return 1;
	}
	else {
		int low = 0, high = szz(L) - 1, res = -1;
		while(low <= high) {
			int mid = (low + high) >> 1;
			vector <int> nh;
			rep(i, szz(L)) if(i != mid) nh.pb(L[i]);
			int ok = 1;
			if(good(nh, mxl - 2) == 0) ok = 0;
			if(ok) res = mid, high = mid - 1;
			else low = mid + 1;
		}
		if(res == -1 || L[res] + 1 > mxl) return 0;
		d[x] = L[res] + 1;
		return 1;
	}
}

void solve() {
	scanf("%d", &n);
	rep(i, n-1) {
		int x, y; scanf("%d%d", &x, &y);
		E[x].pb(y);
		E[y].pb(x);
	}
	
	int odd = 0;
	for(int i=1;i<=n;i++) {
		if(szz(E[i]) & 1) ++odd;
	}
	
	printf("%d ", odd / 2);
	int low = 1, high = n-1, res = -1;
	while(low <= high) {
		int mid = (low + high) >> 1;
		if(dfs(1, -1, mid)) res = mid, high = mid - 1;
		else low = mid + 1;
	}
	printf("%d\n", res);
}

int main(){
	int Tc = 1; // scanf("%d\n", &Tc);
	for(int tc=1;tc<=Tc;tc++){
		solve();
	}
	return 0;
}