Table of contents

• Problem
  • Input
  • Output
• Explanation
• Time complexity
• Solution

# Problem

https://atcoder.jp/contests/arc037/tasks/arc037_b

# Input

$N M$
$u_1 v_1$
$u_2 v_2$
...
$u_M v_M$

• $N$ - The number of vertices
• $M$ - The number of undirected edges
• $u_i v_i$ - Edge which connects $u_i$ and $v_i$

# Output

Report the number of trees.

A tree is either

• 1. A connected graph which has only 1 vertex.
• 2. A connected graph which doesn't have a cycle.

# Explanation

For each vertex, visit it if you haven't visited it yet.
If the vertex doen't have any edge, it's a tree.
If it has edges, visit its neighbor vertices.
If a visited vertex is found, it's a cycle, not a tree.
If any visited vertex is not found to the end, it's a tree.

You visit a vertex just once, so the time complexity is $O(N)$.

# Time complexity

$O(N)$

# Solution

// C++ 14
#include <iostream>
#include <string>
#include <vector>
#include <list>
#include <algorithm>
#include <queue>
#include <stack>
#include <set>
#include <map>
#include <unordered_map>
#include <math.h>

#define ll long long
#define Int int
#define loop(x, start, end) for(Int x = start; x < end; x++)
#define loopdown(x, start, end) for(int x = start; x > end; x--)
#define rep(n) for(int x = 0; x < n; x++)
#define span(a,x,y) a.begin()+x,a.begin()+y
#define span_all(a) a.begin(),a.end()
#define len(x) (x.size())
#define last(x) (*(x.end()-1))

using namespace std;

#define MAX_N 100

Int N, M;
vector<Int> G[MAX_N];
vector<Int> done(MAX_N, false);

void input() {
  Int u,v;
  cin >> N >> M;

  while (cin >> u >> v) {
    u--,v--;
    G[u].push_back(v);
    G[v].push_back(u);
  }
  done.resize(N);
}

bool dfs_isTree(Int u, Int depth, Int parent) {
  done[u] = true;
  if (len(G[u]) == 0) return true;

  bool cycle = false;
  for (auto v: G[u]) {
    if (v == parent) continue;
    if (done[v]) return false;
    cycle |= !dfs_isTree(v, depth + 1, u);
  }
  return !cycle;
}

void solve() {
  Int counter = 0;
  loop(u,0,N) {
    if (done[u]) continue;
    counter += dfs_isTree(u, 0, -1);
  }
  cout << counter << endl;
}

int main(void) {
  input();
  solve();
  return 0;
}