Lemma 17.3

Huffman exhibits optimal substructure.

Proof: Consider the optimal tree T for characters C.

• Let x and y be lowest frequency characters in C.
• Consider the optimal tree T' for C' = C - {x,y} $\cup$ {z}, where f(z) = f(x) + f(y).
• If there is a better tree for C', call it T'', then we could use T'' to build a better original tree by adding in x and y under z.
• The original tree T is optimal, so this is a contradiction.
• Thus T' is the optimal tree for C'.
• Huffman exhibits optimal substructure.

Theorem 17.4

Huffman produces optimal prefix codes.