Binary Trees - 2

Theorem. In a complete binary tree with n vertices and height H,
 
     2^H <= n < 2^H+1.

    Proof.

1. The number of possible vertices in layer k is 2^k .
2. If all of the layers through layer k are full, then the number of vertices in the first k layers is the sum
   1 + 2 + 4 + ... + 2^k = 2^k+1 - 1 .
3. Substitute k = H - 1 : there are 2^H - 1 vertices in the first H - 1 layers.
4. Therefore there are at least this many plus one, i.e., 2^H, vertices in the tree. That is, 2^H <= n .
5. Substitute k = H: the maximum number of vertices is 2^H+1 - 1, which is smaller than 2^H+1 .
6. Therefore, n < 2^H+1.