#include <iostream>
#include <vector>
#include <cstdio>
#include <sstream>
#include <map>
#include <string>
#include <algorithm>
#include <queue>
#include <cmath>
#include <functional>
#include <set>
#include <ctime>
#include <random>
#include <chrono>
#include <cassert>
using namespace std;

namespace {
  using Integer = long long; //__int128;
  template<class T> istream& operator >> (istream& is, vector<T>& vec){for(T& val: vec) is >> val; return is;}
  template<class T> istream& operator ,  (istream& is, T& val){ return is >> val;}
  template<class T> ostream& operator << (ostream& os, const vector<T>& vec){for(int i=0; i<vec.size(); i++) os << vec[i] << (i==vec.size()-1?"":" "); return os;}
  template<class T> ostream& operator ,  (ostream& os, const T& val){ return os << " " << val;}

  template<class H> void print(const H& head){ cout << head; }
  template<class H, class ... T> void print(const H& head, const T& ... tail){ cout << head << " "; print(tail...); }
  template<class ... T> void println(const T& ... values){ print(values...); cout << endl; }

  template<class H> void eprint(const H& head){ cerr << head; }
  template<class H, class ... T> void eprint(const H& head, const T& ... tail){ cerr << head << " "; eprint(tail...); }
  template<class ... T> void eprintln(const T& ... values){ eprint(values...); cerr << endl; }

  class range{
    long long start_, end_, step_;
   public:
    struct range_iterator{
      long long val, step_;
      range_iterator(long long v, long long step) : val(v), step_(step) {}
      long long operator * (){return val;}
      void operator ++ (){val += step_;}
      bool operator != (range_iterator& x){return step_ > 0 ? val < x.val : val > x.val;}
    };
    range(long long len) : start_(0), end_(len), step_(1) {}
    range(long long start, long long end) : start_(start), end_(end), step_(1) {}
    range(long long start, long long end, long long step) : start_(start), end_(end), step_(step) {}
    range_iterator begin(){ return range_iterator(start_, step_); }
    range_iterator   end(){ return range_iterator(  end_, step_); }
  };

  string operator "" _s (const char* str, size_t size){ return move(string(str)); }
  constexpr Integer my_pow(Integer x, Integer k, Integer z=1){return k==0 ? z : k==1 ? z*x : (k&1) ? my_pow(x*x,k>>1,z*x) : my_pow(x*x,k>>1,z);}
  constexpr Integer my_pow_mod(Integer x, Integer k, Integer M, Integer z=1){return k==0 ? z%M : k==1 ? z*x%M : (k&1) ? my_pow_mod(x*x%M,k>>1,M,z*x%M) : my_pow_mod(x*x%M,k>>1,M,z);}
  constexpr unsigned long long operator "" _ten (unsigned long long value){ return my_pow(10,value); }

  inline int k_bit(Integer x, int k){return (x>>k)&1;} //0-indexed

  mt19937 mt(chrono::duration_cast<chrono::nanoseconds>(chrono::steady_clock::now().time_since_epoch()).count());

  template<class T> string join(const vector<T>& v, const string& sep){
    stringstream ss; for(int i=0; i<v.size(); i++){ if(i>0) ss << sep; ss << v[i]; } return ss.str();
  }

  string operator * (string s, int k){ string ret; while(k){ if(k&1) ret += s; s += s; k >>= 1; } return ret; }

}

constexpr long long mod = 9_ten + 7;

//O(v log v)
class centroid_decomposition{
 public:
  const vector<vector<int>>& G;
  vector<vector<int>> H;
  int root;

  centroid_decomposition(const vector<vector<int>>& G_)
   : G(G_), H(G_.size()), subtree(G_.size(), 0), root(-1)
  {
    vector<bool> used(G.size(), false);
    rec(used, 0);
  }

 private:
  vector<int> subtree;
  int get_centroid(vector<bool>& used, int p){
    function<int(int,int)> dfs = [&](int pos, int last){
      int ret = 1;
      for(int nx : G[pos]){
        if(nx == last || used[nx]) continue;
        ret += dfs(nx, pos);
      }
      return subtree[pos] = ret;
    };

    int num_v = dfs(p, p);

    function<int(int,int)> dfs2 = [&](int pos, int last){
      pair<int,int> mx  = {0,-1};
      for(int nx : G[pos]){
        if(nx == last || used[nx]) continue;
        mx = max(mx, pair<int,int>{subtree[nx],nx} );
      }
      if( mx.first <= num_v/2 ) return pos;
      return dfs2(mx.second, pos);
    };

    return dfs2(p, p);
  }

  int rec(vector<bool>& used, int p){
    int r = get_centroid(used, p);
    if(root == -1) root = r;
    used[r] = true;
    for(int nx : G[r]){
      if(!used[nx]){
        H[r].push_back( rec(used, nx) );
      }
    }
    return r;
  }
};

vector<bool> used;

long long get_count (vector<pair<int,int>>& out, vector<pair<int,int>>& in){
  long long ret = 0;
  sort(in.begin(), in.end());
  sort(out.begin(), out.end());
  for(auto p : in){
    ret += lower_bound(out.begin(), out.end(), pair<int,int>{-p.first+1, -10000000}) - 
           lower_bound(out.begin(), out.end(), pair<int,int>{-p.first, -p.first});
  }
  return ret;
}

void dfs(
  string& s,
  vector<vector<int>>&G,
  vector<pair<int,int>>& go_,
  vector<pair<int,int>>& come_,
  int pos, int par, pair<int,int> vo, pair<int,int> vi
){
  int d = (s[pos]=='('?1:-1);
  pair<int,int> vo_ = vo;
  vo_.first  = vo.first + d;
  vo_.second = min(vo.second, vo_.first);

  pair<int,int> vi_ = vo;
  vi_.first  = vi.first + d;
  vi_.second = min(0, vi.second + d);

  if(s[pos] == ')' && vo_.first <= 0)  go_.push_back(vo_);
  if(vi_.second >= 0) come_.push_back(vi_);
  for(int nx : G[pos]){
    if(nx == par || used[nx]) continue;
    dfs(s, G, go_, come_, nx, pos, vo_, vi_);
  }
}

long long solve(vector<vector<int>>& G, vector<vector<int>>& H, string& s, int pos, int last){
  used[pos] = true;

  long long ans = 0;
  for(int nx : H[pos]){
    ans += solve(G,H,s, nx, pos);
  }

  //{sum, min}
  vector<pair<int,int>> go;   // pos -> v
  vector<pair<int,int>> come; // v -> pos

  for(int nx : G[pos]){
    if(nx == last || used[nx]) continue;

    vector<pair<int,int>> go_;   // pos -> v
    vector<pair<int,int>> come_; // v -> pos

    dfs(s,G,go_,come_, nx, pos, {0,0}, {s[pos]=='('?1:-1, s[pos]=='('?0:-1});
    ans -= get_count(go_, come_);
    go.insert(go.end(), go_.begin(), go_.end());
    come.insert(come.end(), come_.begin(), come_.end());
  }


  if(s[pos] == ')')for(auto p : come){
    if(p.first == 0 && p.second >= 0) ans++;
  }
  if(s[pos] == '(') for(auto p : go){
    if(p.first == -1 && p.second+1 >= 0) ans++;
  }

  ans += get_count(go, come);

  used[pos] = false;

  return ans;
}


#include <cassert>
int main(){
  int n;
  cin >> n;
  string s;
  cin >> s;
  vector<vector<int>> G(n);
  for(int i : range(n-1)){
    int u,v;
    //cin >> u,v;
    scanf("%d%d", &u,&v);
    u--; v--;
    G[u].push_back(v);
    G[v].push_back(u);
  }

  centroid_decomposition cd(G);
  auto& H = cd.H;

  used.resize(n, false);
  long long ans = solve(G,H,s, cd.root, -1);

  println(ans);
  return 0;
}
### Prompt

Please provide a cpp coded solution to the problem described below:

Example

Input

2
()
1 2


Output

1

### Response

```cpp
#include <iostream>
#include <vector>
#include <cstdio>
#include <sstream>
#include <map>
#include <string>
#include <algorithm>
#include <queue>
#include <cmath>
#include <functional>
#include <set>
#include <ctime>
#include <random>
#include <chrono>
#include <cassert>
using namespace std;

namespace {
  using Integer = long long; //__int128;
  template<class T> istream& operator >> (istream& is, vector<T>& vec){for(T& val: vec) is >> val; return is;}
  template<class T> istream& operator ,  (istream& is, T& val){ return is >> val;}
  template<class T> ostream& operator << (ostream& os, const vector<T>& vec){for(int i=0; i<vec.size(); i++) os << vec[i] << (i==vec.size()-1?"":" "); return os;}
  template<class T> ostream& operator ,  (ostream& os, const T& val){ return os << " " << val;}

  template<class H> void print(const H& head){ cout << head; }
  template<class H, class ... T> void print(const H& head, const T& ... tail){ cout << head << " "; print(tail...); }
  template<class ... T> void println(const T& ... values){ print(values...); cout << endl; }

  template<class H> void eprint(const H& head){ cerr << head; }
  template<class H, class ... T> void eprint(const H& head, const T& ... tail){ cerr << head << " "; eprint(tail...); }
  template<class ... T> void eprintln(const T& ... values){ eprint(values...); cerr << endl; }

  class range{
    long long start_, end_, step_;
   public:
    struct range_iterator{
      long long val, step_;
      range_iterator(long long v, long long step) : val(v), step_(step) {}
      long long operator * (){return val;}
      void operator ++ (){val += step_;}
      bool operator != (range_iterator& x){return step_ > 0 ? val < x.val : val > x.val;}
    };
    range(long long len) : start_(0), end_(len), step_(1) {}
    range(long long start, long long end) : start_(start), end_(end), step_(1) {}
    range(long long start, long long end, long long step) : start_(start), end_(end), step_(step) {}
    range_iterator begin(){ return range_iterator(start_, step_); }
    range_iterator   end(){ return range_iterator(  end_, step_); }
  };

  string operator "" _s (const char* str, size_t size){ return move(string(str)); }
  constexpr Integer my_pow(Integer x, Integer k, Integer z=1){return k==0 ? z : k==1 ? z*x : (k&1) ? my_pow(x*x,k>>1,z*x) : my_pow(x*x,k>>1,z);}
  constexpr Integer my_pow_mod(Integer x, Integer k, Integer M, Integer z=1){return k==0 ? z%M : k==1 ? z*x%M : (k&1) ? my_pow_mod(x*x%M,k>>1,M,z*x%M) : my_pow_mod(x*x%M,k>>1,M,z);}
  constexpr unsigned long long operator "" _ten (unsigned long long value){ return my_pow(10,value); }

  inline int k_bit(Integer x, int k){return (x>>k)&1;} //0-indexed

  mt19937 mt(chrono::duration_cast<chrono::nanoseconds>(chrono::steady_clock::now().time_since_epoch()).count());

  template<class T> string join(const vector<T>& v, const string& sep){
    stringstream ss; for(int i=0; i<v.size(); i++){ if(i>0) ss << sep; ss << v[i]; } return ss.str();
  }

  string operator * (string s, int k){ string ret; while(k){ if(k&1) ret += s; s += s; k >>= 1; } return ret; }

}

constexpr long long mod = 9_ten + 7;

//O(v log v)
class centroid_decomposition{
 public:
  const vector<vector<int>>& G;
  vector<vector<int>> H;
  int root;

  centroid_decomposition(const vector<vector<int>>& G_)
   : G(G_), H(G_.size()), subtree(G_.size(), 0), root(-1)
  {
    vector<bool> used(G.size(), false);
    rec(used, 0);
  }

 private:
  vector<int> subtree;
  int get_centroid(vector<bool>& used, int p){
    function<int(int,int)> dfs = [&](int pos, int last){
      int ret = 1;
      for(int nx : G[pos]){
        if(nx == last || used[nx]) continue;
        ret += dfs(nx, pos);
      }
      return subtree[pos] = ret;
    };

    int num_v = dfs(p, p);

    function<int(int,int)> dfs2 = [&](int pos, int last){
      pair<int,int> mx  = {0,-1};
      for(int nx : G[pos]){
        if(nx == last || used[nx]) continue;
        mx = max(mx, pair<int,int>{subtree[nx],nx} );
      }
      if( mx.first <= num_v/2 ) return pos;
      return dfs2(mx.second, pos);
    };

    return dfs2(p, p);
  }

  int rec(vector<bool>& used, int p){
    int r = get_centroid(used, p);
    if(root == -1) root = r;
    used[r] = true;
    for(int nx : G[r]){
      if(!used[nx]){
        H[r].push_back( rec(used, nx) );
      }
    }
    return r;
  }
};

vector<bool> used;

long long get_count (vector<pair<int,int>>& out, vector<pair<int,int>>& in){
  long long ret = 0;
  sort(in.begin(), in.end());
  sort(out.begin(), out.end());
  for(auto p : in){
    ret += lower_bound(out.begin(), out.end(), pair<int,int>{-p.first+1, -10000000}) - 
           lower_bound(out.begin(), out.end(), pair<int,int>{-p.first, -p.first});
  }
  return ret;
}

void dfs(
  string& s,
  vector<vector<int>>&G,
  vector<pair<int,int>>& go_,
  vector<pair<int,int>>& come_,
  int pos, int par, pair<int,int> vo, pair<int,int> vi
){
  int d = (s[pos]=='('?1:-1);
  pair<int,int> vo_ = vo;
  vo_.first  = vo.first + d;
  vo_.second = min(vo.second, vo_.first);

  pair<int,int> vi_ = vo;
  vi_.first  = vi.first + d;
  vi_.second = min(0, vi.second + d);

  if(s[pos] == ')' && vo_.first <= 0)  go_.push_back(vo_);
  if(vi_.second >= 0) come_.push_back(vi_);
  for(int nx : G[pos]){
    if(nx == par || used[nx]) continue;
    dfs(s, G, go_, come_, nx, pos, vo_, vi_);
  }
}

long long solve(vector<vector<int>>& G, vector<vector<int>>& H, string& s, int pos, int last){
  used[pos] = true;

  long long ans = 0;
  for(int nx : H[pos]){
    ans += solve(G,H,s, nx, pos);
  }

  //{sum, min}
  vector<pair<int,int>> go;   // pos -> v
  vector<pair<int,int>> come; // v -> pos

  for(int nx : G[pos]){
    if(nx == last || used[nx]) continue;

    vector<pair<int,int>> go_;   // pos -> v
    vector<pair<int,int>> come_; // v -> pos

    dfs(s,G,go_,come_, nx, pos, {0,0}, {s[pos]=='('?1:-1, s[pos]=='('?0:-1});
    ans -= get_count(go_, come_);
    go.insert(go.end(), go_.begin(), go_.end());
    come.insert(come.end(), come_.begin(), come_.end());
  }


  if(s[pos] == ')')for(auto p : come){
    if(p.first == 0 && p.second >= 0) ans++;
  }
  if(s[pos] == '(') for(auto p : go){
    if(p.first == -1 && p.second+1 >= 0) ans++;
  }

  ans += get_count(go, come);

  used[pos] = false;

  return ans;
}


#include <cassert>
int main(){
  int n;
  cin >> n;
  string s;
  cin >> s;
  vector<vector<int>> G(n);
  for(int i : range(n-1)){
    int u,v;
    //cin >> u,v;
    scanf("%d%d", &u,&v);
    u--; v--;
    G[u].push_back(v);
    G[v].push_back(u);
  }

  centroid_decomposition cd(G);
  auto& H = cd.H;

  used.resize(n, false);
  long long ans = solve(G,H,s, cd.root, -1);

  println(ans);
  return 0;
}
```