Consider the string abbccddeee. Each letter in the string must be assigned a binary code satisfying the following properties: 1. For any two letters, the code assigned to one letter must not be a prefix of the code assigned to the other letter. 2. For any two letters of the same frequency, the letter which occurs earlier in the dictionary order is assigned a code whose length is at most the length of the code assigned to the other letter. Among the set of all binary code assignments which satisfy the above two properties, what is the minimum length of the encoded string?

First, calculate the frequency of each character in the string abbccddeee:

• $a: 1$

• $b: 2$

• $c: 2$

• $d: 2$

• $e: 3$

Property 1 requires the binary code to be prefix-free, which can be represented as a binary tree with characters as leaves.

Property 2 states that for letters with identical frequencies, earlier letters in alphabetical order must have a code length less than or equal to later letters. The letters with the same frequency are $b, c, d$ (each having frequency $2$). Since the alphabetical order is $b < c < d$, their code lengths must satisfy:

To minimize the total encoded length:

• An optimal prefix code tree has leaf depths of either $2$ or $3$.

• Specifically, we can have three leaves at depth $2$ and two leaves at depth $3$, satisfying Kraft's inequality:

To minimize the weighted path length while satisfying $\text{length}(b) \le \text{length}(c) \le \text{length}(d)$:

• Assign length $2$ to the highest frequency symbol $e$.

• Assign length $2$ to $b$ and $c$, and length $3$ to $d$, which satisfies $\text{length}(b) = 2 \le \text{length}(c) = 2 \le \text{length}(d) = 3$.

• Assign length $3$ to $a$.

Now, calculate the total encoded string length:

Thus, the minimum length of the encoded string is 23 (Option B).