Question 36

The string is abbccddeee, with character counts:

• $a: 1$
• $b: 2$
• $c: 2$
• $d: 2$
• $e: 3$
  Total length of the string is $1 + 2 + 2 + 2 + 3 = 10$ characters.

Constraint 1 requires a prefix code (valid binary tree leaf assignment). Constraint 2 requires that for letters of equal frequency, earlier dictionary letters must have codeword length $\le$ later letters. Specifically, for the equal-frequency letters $b, c, d$ (all frequency 2):

Consider a standard Huffman tree without constraint 2: grouping $a(1)+b(2)=3$, then $c(2)+d(2)=4$, then $e(3)+ab(3)=6$, then $cd(4)+eab(6)=10$. This assigns length 2 to $c, d, e$ and length 3 to $a, b$. However, this gives $\text{length}(b) = 3 > \text{length}(c) = 2$, violating constraint 2 because $b$ occurs before $c$ in dictionary order.

To satisfy $\text{length}(b) \le \text{length}(c) \le \text{length}(d)$, consider binary tree structures with 5 leaves:
Using codeword lengths $\{2, 2, 2, 3, 3\}$ (which satisfies Kraft's inequality: $2^{-2} + 2^{-2} + 2^{-2} + 2^{-3} + 2^{-3} = 1$):

• Assign length 2 to the highest frequency letter $e$ (frequency 3).
• Among $b, c, d$, to satisfy constraint 2, assign length 2 to $b$ and $c$, and length 3 to $d$:

• The remaining leaf of length 3 is assigned to $a$ (frequency 1).

Calculate the total encoded length:

This satisfies both conditions and achieves the minimum possible length, matching option B.