G - Vertex Deletion

Score : $600$ points

Problem Statement

Given is a tree with $N$ vertices. The vertices are numbered $1,2,\ldots,N$, and the $i$-th edge $(1 \leq i \leq N-1)$ connects Vertex $u_i$ and Vertex $v_i$.

Find the number of integers $i$ $(1 \leq i \leq N)$ that satisfy the following condition.

• The size of the maximum matching of the graph obtained by deleting Vertex $i$ and all incident edges from the tree is equal to the size of the maximum matching of the original tree.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq u_i < v_i \leq N$
• The given graph is a tree.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$u_1$ $v_1$

$u_2$ $v_2$

$\vdots$

$u_{N-1}$ $v_{N-1}$

Output

Print the answer.

3
1 2
2 3

2

The size of the maximum matching of the original tree is $1$.

The size of the maximum matching of the graph obtained by deleting Vertex $1$ and all incident edges from the tree is $1$.

The size of the maximum matching of the graph obtained by deleting Vertex $2$ and all incident edges from the tree is $0$.

The size of the maximum matching of the graph obtained by deleting Vertex $3$ and all incident edges from the tree is $1$.

Thus, two integers $i=1,3$ satisfy the condition, so we should print $2$.

2
1 2

0

6
2 5
3 5
1 4
4 5
4 6

4