title: proof of time complexity of union find with path compression
body: <p>The Wikipedia page: <a href="https://en.wikipedia.org/wiki/Proof_of_O(log*n)_time_complexity_of_union%E2%80%93find" rel="nofollow noreferrer">wikipedia page</a> states that </p>
  
  <blockquote>
    <p>If m operations, either Union or Find, are applied to n elements, the total run time is O(m log*n).</p>
  </blockquote>
  
  <p>The detailed analysis arrived at this result is :</p>
  
  <p><a href="https://i.stack.imgur.com/zbBST.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zbBST.png" alt="enter image description here"></a></p>
  
  <p>My questions are:</p>
  
  <ol>
  <li>Shouldn't the sum be <code>(m+n)log*n</code> instead of <code>mlog*n</code>? </li>
  <li>Are the average time complexity for 1000 Find operations the same as the time complexity of each individual Find? </li>
  </ol>