<p>Given a string <code>formula</code> representing a chemical formula, return <em>the count of each atom</em>.</p>

<p>The atomic element always starts with an uppercase character, then zero or more lowercase letters, representing the name.</p>

<p>One or more digits representing that element&#39;s count may follow if the count is greater than <code>1</code>. If the count is <code>1</code>, no digits will follow.</p>

<ul>
	<li>For example, <code>&quot;H2O&quot;</code> and <code>&quot;H2O2&quot;</code> are possible, but <code>&quot;H1O2&quot;</code> is impossible.</li>
</ul>

<p>Two formulas are concatenated together to produce another formula.</p>

<ul>
	<li>For example, <code>&quot;H2O2He3Mg4&quot;</code> is also a formula.</li>
</ul>

<p>A formula placed in parentheses, and a count (optionally added) is also a formula.</p>

<ul>
	<li>For example, <code>&quot;(H2O2)&quot;</code> and <code>&quot;(H2O2)3&quot;</code> are formulas.</li>
</ul>

<p>Return the count of all elements as a string in the following form: the first name (in sorted order), followed by its count (if that count is more than <code>1</code>), followed by the second name (in sorted order), followed by its count (if that count is more than <code>1</code>), and so on.</p>

<p>The test cases are generated so that all the values in the output fit in a <strong>32-bit</strong> integer.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> formula = &quot;H2O&quot;
<strong>Output:</strong> &quot;H2O&quot;
<strong>Explanation:</strong> The count of elements are {&#39;H&#39;: 2, &#39;O&#39;: 1}.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> formula = &quot;Mg(OH)2&quot;
<strong>Output:</strong> &quot;H2MgO2&quot;
<strong>Explanation:</strong> The count of elements are {&#39;H&#39;: 2, &#39;Mg&#39;: 1, &#39;O&#39;: 2}.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> formula = &quot;K4(ON(SO3)2)2&quot;
<strong>Output:</strong> &quot;K4N2O14S4&quot;
<strong>Explanation:</strong> The count of elements are {&#39;K&#39;: 4, &#39;N&#39;: 2, &#39;O&#39;: 14, &#39;S&#39;: 4}.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= formula.length &lt;= 1000</code></li>
	<li><code>formula</code> consists of English letters, digits, <code>&#39;(&#39;</code>, and <code>&#39;)&#39;</code>.</li>
	<li><code>formula</code> is always valid.</li>
</ul>

class Solution {
 public:
  string countOfAtoms(string formula) {
    string ans;
    int i = 0;

    for (const auto& [elem, freq] : parse(formula, i)) {
      ans += elem;
      if (freq > 1)
        ans += to_string(freq);
    }

    return ans;
  }

 private:
  map<string, int> parse(const string& s, int& i) {
    map<string, int> count;

    while (i < s.length())
      if (s[i] == '(') {
        for (const auto& [elem, freq] : parse(s, ++i))
          count[elem] += freq;
      } else if (s[i] == ')') {
        const int num = getNum(s, ++i);
        for (auto&& [_, freq] : count)
          freq *= num;
        return count;  
      } else {         
        const string& elem = getElem(s, i);
        const int num = getNum(s, i);
        count[elem] += num;
      }

    return count;
  }

  string getElem(const string& s, int& i) {
    const int elemStart = i++;  // `s[elemStart]` is uppercased.
    while (i < s.length() && islower(s[i]))
      ++i;
    return s.substr(elemStart, i - elemStart);
  }

  int getNum(const string& s, int& i) {
    const int numStart = i;
    while (i < s.length() && isdigit(s[i]))
      ++i;
    const string& numString = s.substr(numStart, i - numStart);
    return numString.empty() ? 1 : stoi(numString);
  }
};
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