200883: [AtCoder]ARC088 F - Christmas Tree

Problem Statement

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.

Sample Input 1

7
1 2
2 3
2 4
4 5
4 6
6 7

Sample Output 1

3 2

We can make a Christmas Tree as shown in the figure below:

Sample Input 2

8
1 2
2 3
3 4
4 5
5 6
5 7
5 8

Sample Output 2

2 5

Sample Input 3

10
1 2
2 3
3 4
2 5
6 5
6 7
7 8
5 9
10 5

Sample Output 3

3 4