title: Largest and smallest number of internal nodes in red-black tree?
body: <p>The smallest number of internal nodes in a red-black tree with black height of k is 2<sup>k</sup>-1 which is one in the following image:</p>
  
  <p><img src="https://i.stack.imgur.com/IxXnq.png" alt="enter image description here"></p>
  
  <p>The largest number of internal nodes with black height of k is 2<sup>2k</sup>-1 which, if the black height is 2, should be 2<sup>4</sup> - 1 = 15.  However, consider this image:</p>
  
  <p><img src="https://i.stack.imgur.com/kVB9j.png" alt="enter image description here"></p>
  
  <p>The number of internal nodes is 7. What am I doing wrong?</p>