1. **Problem Statement:** Determine which of the given sets of codewords $p_1$ to $p_6$ are prefix codes. A prefix code means no codeword is a prefix of another codeword in the same set. 2. **Definition:** A prefix code is a set of codewords where no codeword is the prefix of any other codeword. 3. **Check $p_1 = \{00, 10, 010, 011, 1010, 1100, 1101\}$:** - Check if any codeword is prefix of another. - $00$ is prefix of $010$ and $011$? No, $00$ vs $010$ starts with $0$ but $010$ starts with $0$ then $1$, so $00$ is prefix of $010$? $00$ vs $010$: $00$ vs $01$ no, so no. - $10$ is prefix of $1010$? Yes, $10$ is prefix of $1010$. Since $10$ is prefix of $1010$, $p_1$ is **not** a prefix code. 4. **Check $p_2 = \{00, 10, 010, 011, 111, 1100, 1101\}$:** - Check prefixes: - $00$ vs $010$? $00$ is prefix of $010$? $00$ vs $01$ no. - $10$ vs $1100$? $10$ vs $11$ no. - $1100$ vs $1101$? $1100$ is prefix of $1101$? No, they differ at last bit. No codeword is prefix of another, so $p_2$ **is** a prefix code. 5. **Check $p_3 = \{111, 1100, 1010, 1111, 010\}$:** - $111$ is prefix of $1111$? Yes, $111$ is prefix of $1111$. So $p_3$ is **not** a prefix code. 6. **Check $p_4 = \{000, 001, 01, 10, 11\}$:** - $000$ and $001$ share prefix $00$, but neither is prefix of the other. - $01$ is prefix of $000$ or $001$? No. - $01$ is prefix of $10$ or $11$? No. No codeword is prefix of another, so $p_4$ **is** a prefix code. 7. **Check $p_5 = \{01, 0, 101, 10, 1\}$:** - $0$ is prefix of $01$ and $10$? $0$ is prefix of $01$ yes. So $p_5$ is **not** a prefix code. 8. **Check $p_6 = \{1, 00, 01, 000, 0001\}$:** - $00$ is prefix of $000$ and $0001$ yes. So $p_6$ is **not** a prefix code. **Summary:** - Prefix codes are $p_2$ and $p_4$. --- **Binary Tree for $p_2$:** - Root node splits into 0 and 1. - For codewords starting with 0: $00$, $010$, $011$ - For codewords starting with 1: $10$, $111$, $1100$, $1101$ **Binary Tree for $p_4$:** - Root splits into 0 and 1. - Left subtree: $000$, $001$, $01$ - Right subtree: $10$, $11$ Due to text limitations, the binary tree is described conceptually rather than drawn.