Problem:
Score : 900 points

Problem Statement

We have a tree with N vertices.
The vertices are numbered 0 through N - 1, and the i-th edge (0 ≤ i < N - 1) comnnects Vertex a_i and b_i.
For each pair of vertices u and v (0 ≤ u, v < N), we define the distance d(u, v) as the number of edges in the path u-v.

It is expected that one of the vertices will be invaded by aliens from outer space.
Snuke wants to immediately identify that vertex when the invasion happens.
To do so, he has decided to install an antenna on some vertices.

First, he decides the number of antennas, K (1 ≤ K ≤ N).
Then, he chooses K different vertices, x_0, x_1, ..., x_{K - 1}, on which he installs Antenna 0, 1, ..., K - 1, respectively.
If Vertex v is invaded by aliens, Antenna k (0 ≤ k < K) will output the distance d(x_k, v).
Based on these K outputs, Snuke will identify the vertex that is invaded.
Thus, in order to identify the invaded vertex no matter which one is invaded, the following condition must hold:

For each vertex u (0 ≤ u < N), consider the vector (d(x_0, u), ..., d(x_{K - 1}, u)). These N vectors are distinct.

Find the minumum value of K, the number of antennas, when the condition is satisfied.

Constraints

2 ≤ N ≤ 10^5

0 ≤ a_i, b_i < N

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N
a_0 b_0
a_1 b_1
:
a_{N - 2} b_{N - 2}

Output

Print the minumum value of K, the number of antennas, when the condition is satisfied.

Sample Input 1

5
0 1
0 2
0 3
3 4

Sample Output 1

2

For example, install an antenna on Vertex 1 and 3.
Then, the following five vectors are distinct:

(d(1, 0), d(3, 0)) = (1, 1)

(d(1, 1), d(3, 1)) = (0, 2)

(d(1, 2), d(3, 2)) = (2, 2)

(d(1, 3), d(3, 3)) = (2, 0)

(d(1, 4), d(3, 4)) = (3, 1)

Sample Input 2

2
0 1

Sample Output 2

1

For example, install an antenna on Vertex 0.

Sample Input 3

10
2 8
6 0
4 1
7 6
2 3
8 6
6 9
2 4
5 8

Sample Output 3

3

For example, install an antenna on Vertex 0, 4, 9.

Solution in Rust (1.42.0) (judge verdict: Accepted)

pub trait Zero: PartialEq + Sized {
    fn zero() -> Self;
    #[inline]
    fn is_zero(&self) -> bool {
        self == &Self::zero()
    }
}
pub trait One: PartialEq + Sized {
    fn one() -> Self;
    #[inline]
    fn is_one(&self) -> bool {
        self == &Self::one()
    }
}
macro_rules ! zero_one_impls {($ ({$ Trait : ident $ method : ident $ ($ t : ty ) *, $ e : expr } ) * ) => {$ ($ (impl $ Trait for $ t {# [inline ] fn $ method () -> Self {$ e } } ) * ) * } ; }
zero_one_impls ! ({Zero zero u8 u16 u32 u64 usize i8 i16 i32 i64 isize u128 i128 , 0 } {Zero zero f32 f64 , 0. } {One one u8 u16 u32 u64 usize i8 i16 i32 i64 isize u128 i128 , 1 } {One one f32 f64 , 1. } );
pub trait IterScan: Sized {
    type Output;
    fn scan<'a, I: Iterator<Item = &'a str>>(iter: &mut I) -> Option<Self::Output>;
}
pub trait MarkedIterScan: Sized {
    type Output;
    fn mscan<'a, I: Iterator<Item = &'a str>>(self, iter: &mut I) -> Option<Self::Output>;
}
#[derive(Clone, Debug)]
pub struct Scanner<'a> {
    iter: std::str::SplitAsciiWhitespace<'a>,
}
mod scanner_impls {
    use super::*;
    impl<'a> Scanner<'a> {
        #[inline]
        pub fn new(s: &'a str) -> Self {
            let iter = s.split_ascii_whitespace();
            Self { iter }
        }
        #[inline]
        pub fn scan<T: IterScan>(&mut self) -> <T as IterScan>::Output {
            <T as IterScan>::scan(&mut self.iter).expect("scan error")
        }
        #[inline]
        pub fn mscan<T: MarkedIterScan>(&mut self, marker: T) -> <T as MarkedIterScan>::Output {
            marker.mscan(&mut self.iter).expect("scan error")
        }
        #[inline]
        pub fn scan_vec<T: IterScan>(&mut self, size: usize) -> Vec<<T as IterScan>::Output> {
            (0..size)
                .map(|_| <T as IterScan>::scan(&mut self.iter).expect("scan error"))
                .collect()
        }
        #[inline]
        pub fn iter<'b, T: IterScan>(&'b mut self) -> ScannerIter<'a, 'b, T> {
            ScannerIter {
                inner: self,
                _marker: std::marker::PhantomData,
            }
        }
    }
    macro_rules ! iter_scan_impls {($ ($ t : ty ) * ) => {$ (impl IterScan for $ t {type Output = Self ; # [inline ] fn scan <'a , I : Iterator < Item = &'a str >> (iter : & mut I ) -> Option < Self > {iter . next () ?. parse ::<$ t > () . ok () } } ) * } ; }
    iter_scan_impls ! (char u8 u16 u32 u64 usize i8 i16 i32 i64 isize f32 f64 u128 i128 String );
    macro_rules ! iter_scan_tuple_impl {($ ($ T : ident ) * ) => {impl <$ ($ T : IterScan ) ,*> IterScan for ($ ($ T , ) * ) {type Output = ($ (<$ T as IterScan >:: Output , ) * ) ; # [inline ] fn scan <'a , It : Iterator < Item = &'a str >> (_iter : & mut It ) -> Option < Self :: Output > {Some (($ (<$ T as IterScan >:: scan (_iter ) ?, ) * ) ) } } } ; }
    iter_scan_tuple_impl!();
    iter_scan_tuple_impl!(A);
    iter_scan_tuple_impl ! (A B );
    iter_scan_tuple_impl ! (A B C );
    iter_scan_tuple_impl ! (A B C D );
    iter_scan_tuple_impl ! (A B C D E );
    iter_scan_tuple_impl ! (A B C D E F );
    iter_scan_tuple_impl ! (A B C D E F G );
    iter_scan_tuple_impl ! (A B C D E F G H );
    iter_scan_tuple_impl ! (A B C D E F G H I );
    iter_scan_tuple_impl ! (A B C D E F G H I J );
    iter_scan_tuple_impl ! (A B C D E F G H I J K );
    pub struct ScannerIter<'a, 'b, T> {
        inner: &'b mut Scanner<'a>,
        _marker: std::marker::PhantomData<fn() -> T>,
    }
    impl<'a, 'b, T: IterScan> Iterator for ScannerIter<'a, 'b, T> {
        type Item = <T as IterScan>::Output;
        #[inline]
        fn next(&mut self) -> Option<Self::Item> {
            <T as IterScan>::scan(&mut self.inner.iter)
        }
    }
}
pub mod marker {
    use super::*;
    use std::{iter::FromIterator, marker::PhantomData};
    #[derive(Debug, Copy, Clone)]
    pub struct Usize1;
    impl IterScan for Usize1 {
        type Output = usize;
        #[inline]
        fn scan<'a, I: Iterator<Item = &'a str>>(iter: &mut I) -> Option<Self::Output> {
            Some(<usize as IterScan>::scan(iter)?.checked_sub(1)?)
        }
    }
    #[derive(Debug, Copy, Clone)]
    pub struct Chars;
    impl IterScan for Chars {
        type Output = Vec<char>;
        #[inline]
        fn scan<'a, I: Iterator<Item = &'a str>>(iter: &mut I) -> Option<Self::Output> {
            Some(iter.next()?.chars().collect())
        }
    }
    #[derive(Debug, Copy, Clone)]
    pub struct CharsWithBase(pub char);
    impl MarkedIterScan for CharsWithBase {
        type Output = Vec<usize>;
        #[inline]
        fn mscan<'a, I: Iterator<Item = &'a str>>(self, iter: &mut I) -> Option<Self::Output> {
            Some(
                iter.next()?
                    .chars()
                    .map(|c| (c as u8 - self.0 as u8) as usize)
                    .collect(),
            )
        }
    }
    #[derive(Debug, Copy, Clone)]
    pub struct Collect<T: IterScan, B: FromIterator<<T as IterScan>::Output>> {
        size: usize,
        _marker: PhantomData<fn() -> (T, B)>,
    }
    impl<T: IterScan, B: FromIterator<<T as IterScan>::Output>> Collect<T, B> {
        pub fn new(size: usize) -> Self {
            Self {
                size,
                _marker: PhantomData,
            }
        }
    }
    impl<T: IterScan, B: FromIterator<<T as IterScan>::Output>> MarkedIterScan for Collect<T, B> {
        type Output = B;
        #[inline]
        fn mscan<'a, I: Iterator<Item = &'a str>>(self, iter: &mut I) -> Option<Self::Output> {
            Some(
                (0..self.size)
                    .map(|_| <T as IterScan>::scan(iter).expect("scan error"))
                    .collect::<B>(),
            )
        }
    }
}
#[macro_export]
macro_rules ! min {($ e : expr ) => {$ e } ; ($ e : expr , $ ($ es : expr ) ,+ ) => {std :: cmp :: min ($ e , min ! ($ ($ es ) ,+ ) ) } ; }
#[macro_export]
macro_rules ! chmin {($ dst : expr , $ ($ src : expr ) ,+ ) => {{let x = std :: cmp :: min ($ dst , min ! ($ ($ src ) ,+ ) ) ; $ dst = x ; } } ; }
#[macro_export]
macro_rules ! max {($ e : expr ) => {$ e } ; ($ e : expr , $ ($ es : expr ) ,+ ) => {std :: cmp :: max ($ e , max ! ($ ($ es ) ,+ ) ) } ; }
#[macro_export]
macro_rules ! chmax {($ dst : expr , $ ($ src : expr ) ,+ ) => {{let x = std :: cmp :: max ($ dst , max ! ($ ($ src ) ,+ ) ) ; $ dst = x ; } } ; }
pub use sparse_graph::{
    Adjacency, BidirectionalGraphScanner, BidirectionalSparseGraph, DirectedGraphScanner,
    DirectedSparseGraph, SparseGraph, TreeGraphScanner, UndirectedGraphScanner,
    UndirectedSparseGraph,
};
pub mod sparse_graph {
    use super::*;
    use std::{iter, marker::PhantomData, ops, slice};
    type Marker<T> = PhantomData<fn() -> T>;
    #[derive(Clone, Copy, Debug, Default, Eq, PartialEq, Ord, PartialOrd, Hash)]
    pub struct DirectedEdge;
    #[derive(Clone, Copy, Debug, Default, Eq, PartialEq, Ord, PartialOrd, Hash)]
    pub struct UndirectedEdge;
    #[derive(Clone, Copy, Debug, Default, Eq, PartialEq, Ord, PartialOrd, Hash)]
    pub struct BidirectionalEdge;
    #[derive(Clone, Copy, Debug, Default, Eq, PartialEq, Ord, PartialOrd, Hash)]
    pub struct Adjacency {
        pub id: usize,
        pub to: usize,
    }
    impl Adjacency {
        pub fn new(id: usize, to: usize) -> Adjacency {
            Adjacency { id, to }
        }
    }
    /// Static Sparse Graph represented as Compressed Sparse Row.
    #[derive(Debug, Clone)]
    pub struct SparseGraph<D> {
        vsize: usize,
        pub start: Vec<usize>,
        pub elist: Vec<Adjacency>,
        pub edges: Vec<(usize, usize)>,
        _marker: Marker<D>,
    }
    impl<D> SparseGraph<D> {
        /// Return the number of vertices.
        pub fn vertices_size(&self) -> usize {
            self.vsize
        }
        /// Return the number of edges.
        pub fn edges_size(&self) -> usize {
            self.edges.len()
        }
        /// Return an iterator over graph vertices.
        pub fn vertices(&self) -> ops::Range<usize> {
            0..self.vertices_size()
        }
        /// Return a slice of adjacency vertices.
        pub fn adjacencies(&self, v: usize) -> slice::Iter<'_, Adjacency> {
            self.elist[self.start[v]..self.start[v + 1]].iter()
        }
    }
    pub trait SparseGraphConstruction: Sized {
        fn construct_graph(vsize: usize, edges: Vec<(usize, usize)>) -> SparseGraph<Self>;
    }
    impl<D: SparseGraphConstruction> SparseGraph<D> {
        /// Construct graph from edges.
        pub fn from_edges(vsize: usize, edges: Vec<(usize, usize)>) -> Self {
            D::construct_graph(vsize, edges)
        }
    }
    impl SparseGraphConstruction for DirectedEdge {
        fn construct_graph(vsize: usize, edges: Vec<(usize, usize)>) -> SparseGraph<Self> {
            let mut start: Vec<_> = iter::repeat(0).take(vsize + 1).collect();
            let mut elist = Vec::with_capacity(edges.len());
            unsafe { elist.set_len(edges.len()) }
            for (from, _) in edges.iter().cloned() {
                start[from] += 1;
            }
            for i in 1..=vsize {
                start[i] += start[i - 1];
            }
            for (id, (from, to)) in edges.iter().cloned().enumerate() {
                start[from] -= 1;
                elist[start[from]] = Adjacency::new(id, to);
            }
            SparseGraph {
                vsize,
                start,
                elist,
                edges,
                _marker: PhantomData,
            }
        }
    }
    impl SparseGraphConstruction for UndirectedEdge {
        fn construct_graph(vsize: usize, edges: Vec<(usize, usize)>) -> SparseGraph<Self> {
            let mut start: Vec<_> = iter::repeat(0).take(vsize + 1).collect();
            let mut elist = Vec::with_capacity(edges.len() * 2);
            unsafe { elist.set_len(edges.len() * 2) }
            for (from, to) in edges.iter().cloned() {
                start[to] += 1;
                start[from] += 1;
            }
            for i in 1..=vsize {
                start[i] += start[i - 1];
            }
            for (id, (from, to)) in edges.iter().cloned().enumerate() {
                start[from] -= 1;
                elist[start[from]] = Adjacency::new(id, to);
                start[to] -= 1;
                elist[start[to]] = Adjacency::new(id, from);
            }
            SparseGraph {
                vsize,
                start,
                elist,
                edges,
                _marker: PhantomData,
            }
        }
    }
    impl SparseGraphConstruction for BidirectionalEdge {
        fn construct_graph(vsize: usize, edges: Vec<(usize, usize)>) -> SparseGraph<Self> {
            let mut start: Vec<_> = iter::repeat(0).take(vsize + 1).collect();
            let mut elist = Vec::with_capacity(edges.len() * 2);
            unsafe { elist.set_len(edges.len() * 2) }
            for (from, to) in edges.iter().cloned() {
                start[to] += 1;
                start[from] += 1;
            }
            for i in 1..=vsize {
                start[i] += start[i - 1];
            }
            for (id, (from, to)) in edges.iter().cloned().enumerate() {
                start[from] -= 1;
                elist[start[from]] = Adjacency::new(id * 2, to);
                start[to] -= 1;
                elist[start[to]] = Adjacency::new(id * 2 + 1, from);
            }
            SparseGraph {
                vsize,
                start,
                elist,
                edges,
                _marker: PhantomData,
            }
        }
    }
    pub type DirectedSparseGraph = SparseGraph<DirectedEdge>;
    pub type UndirectedSparseGraph = SparseGraph<UndirectedEdge>;
    pub type BidirectionalSparseGraph = SparseGraph<BidirectionalEdge>;
    pub struct SparseGraphScanner<U: IterScan<Output = usize>, T: IterScan, D> {
        vsize: usize,
        esize: usize,
        _marker: Marker<(U, T, D)>,
    }
    impl<U: IterScan<Output = usize>, T: IterScan, D> SparseGraphScanner<U, T, D> {
        pub fn new(vsize: usize, esize: usize) -> Self {
            Self {
                vsize,
                esize,
                _marker: PhantomData,
            }
        }
    }
    impl<U: IterScan<Output = usize>, T: IterScan, D: SparseGraphConstruction> MarkedIterScan
        for SparseGraphScanner<U, T, D>
    {
        type Output = (SparseGraph<D>, Vec<<T as IterScan>::Output>);
        fn mscan<'a, I: Iterator<Item = &'a str>>(self, iter: &mut I) -> Option<Self::Output> {
            let mut edges = Vec::with_capacity(self.esize);
            let mut rest = Vec::with_capacity(self.esize);
            for _ in 0..self.esize {
                edges.push((U::scan(iter)?, U::scan(iter)?));
                rest.push(T::scan(iter)?);
            }
            let graph = SparseGraph::from_edges(self.vsize, edges);
            Some((graph, rest))
        }
    }
    pub type DirectedGraphScanner<U, T> = SparseGraphScanner<U, T, DirectedEdge>;
    pub type UndirectedGraphScanner<U, T> = SparseGraphScanner<U, T, UndirectedEdge>;
    pub type BidirectionalGraphScanner<U, T> = SparseGraphScanner<U, T, BidirectionalEdge>;
    pub struct TreeGraphScanner<U: IterScan<Output = usize>, T: IterScan> {
        vsize: usize,
        _marker: Marker<(U, T)>,
    }
    impl<U: IterScan<Output = usize>, T: IterScan> TreeGraphScanner<U, T> {
        pub fn new(vsize: usize) -> Self {
            Self {
                vsize,
                _marker: PhantomData,
            }
        }
    }
    impl<U: IterScan<Output = usize>, T: IterScan> MarkedIterScan for TreeGraphScanner<U, T> {
        type Output = (UndirectedSparseGraph, Vec<<T as IterScan>::Output>);
        fn mscan<'a, I: Iterator<Item = &'a str>>(self, iter: &mut I) -> Option<Self::Output> {
            UndirectedGraphScanner::<U, T>::new(self.vsize, self.vsize - 1).mscan(iter)
        }
    }
}
fn main() {
    #![allow(unused_imports, unused_macros)]
    use std::io::{stdin, stdout, BufWriter, Read as _, Write as _};
    let mut _in_buf = Vec::new();
    stdin().read_to_end(&mut _in_buf).expect("io error");
    let _in_buf = unsafe { String::from_utf8_unchecked(_in_buf) };
    let mut scanner = Scanner::new(&_in_buf);
    macro_rules ! scan {() => {scan ! (usize ) } ; (($ ($ t : tt ) ,* ) ) => {($ (scan ! ($ t ) ) ,* ) } ; ([$ t : tt ; $ len : expr ] ) => {(0 ..$ len ) . map (| _ | scan ! ($ t ) ) . collect ::< Vec < _ >> () } ; ([$ t : ty ; $ len : expr ] ) => {scanner . scan_vec ::<$ t > ($ len ) } ; ([$ t : ty ] ) => {scanner . iter ::<$ t > () } ; ({$ e : expr } ) => {scanner . mscan ($ e ) } ; ($ t : ty ) => {scanner . scan ::<$ t > () } ; }
    let _out = stdout();
    let mut _out = BufWriter::new(_out.lock());
    macro_rules ! print {($ ($ arg : tt ) * ) => (:: std :: write ! (_out , $ ($ arg ) * ) . expect ("io error" ) ) }
    macro_rules ! println {($ ($ arg : tt ) * ) => (:: std :: writeln ! (_out , $ ($ arg ) * ) . expect ("io error" ) ) }
    macro_rules! echo {
        ($ iter : expr ) => {
            echo!($iter, '\n')
        };
        ($ iter : expr , $ sep : expr ) => {
            let mut iter = $iter.into_iter();
            if let Some(item) = iter.next() {
                print!("{}", item);
            }
            for item in iter {
                print!("{}{}", $sep, item);
            }
            println!();
        };
    }
    let n = scan!();
    let (g, _) = scan!({ TreeGraphScanner::<usize, ()>::new(n) });
    let degmx = g
        .vertices()
        .map(|u| g.adjacencies(u).len())
        .max()
        .unwrap_or_default();
    if degmx <= 2 {
        println!("1");
    } else {
        println!(
            "{}",
            g.dfs(
                g.vertices()
                    .max_by_key(|&u| g.adjacencies(u).len())
                    .unwrap_or_default(),
                n,
            )
        );
    }
}
impl UndirectedSparseGraph {
    fn dfs(&self, u: usize, p: usize) -> usize {
        let mut acc = 0;
        let mut z = 0;
        for a in self.adjacencies(u).filter(|a| a.to != p) {
            let x = self.dfs(a.to, u);
            acc += x;
            z += (x == 0) as usize;
        }
        acc + z.saturating_sub(1)
    }
}