Example:
Note:
I.e., every level is completely full of nodes
Proof:
2^{0} = 1
So:
Depth d # nodes at depth d # of child nodes -------------------------------------------------------------- 0 1 = 2^{0} 2 (each node has 2 children) 1 2 = 2^{1} 4 (each node has 2 children) 2 4 = 2^{0} 8 (each node has 2 children) ...
I.e.:
Therefore:
# nodes at depth d = 2^{d}
See:
# nodes = 2^{0} + 2^{1} + ... 2^{h} = 2^{h+1} − 1
S = 1 + 2 + 2^{2} + 2^{3} + ... + 2^{h} 2xS = 2 + 2^{2} + 2^{3} + ... + 2^{h} + 2^{h+1} - (subtract) ------------------------------------------------------------ 2xS - S = 2^{h+1} - 1 <==> S = 2^{h+1} - 1