1. **Stating the problem:** We need to find the lexicographic (infix), postfix, and prefix notations for the expression represented by the given binary tree: $$\frac{(-3 \times -5)}{(5 - 3 + (-3 \times -8) + (6 - 2 \times (-1 + 3)))}$$ 2. **Understanding the tree and notations:** - **Lexicographic (infix):** The usual way of writing expressions with operators between operands. - **Postfix (Reverse Polish Notation):** Operators come after their operands. - **Prefix (Polish Notation):** Operators come before their operands. 3. **Extracting the expression from the tree:** - Left subtree of root "/": $(-3 \times -5)$ - Right subtree of root "/": $(5 - 3 + (-3 \times -8) + (6 - 2 \times (-1 + 3)))$ 4. **Lexicographic (infix) notation:** $$\frac{(-3 \times -5)}{(5 - 3 + (-3 \times -8) + (6 - 2 \times (-1 + 3)))}$$ 5. **Postfix notation:** - Left subtree: $-3\ -5\ \times$ - Right subtree: - Inner left: $5\ 3\ -$ - Inner right: $-3\ -8\ \times$ - Sum of these two: $5\ 3\ -\ -3\ -8\ \times\ +$ - Rightmost subtree: - $1\ 3\ -$ - $2\ (1\ 3\ -)\ \times$ - $6\ (2\ (1\ 3\ -)\ \times)\ -$ - Sum all right subtree parts: $$5\ 3\ -\ -3\ -8\ \times\ +\ 6\ 2\ 1\ 3\ -\ \times\ -\ +$$ - Full postfix: $$-3\ -5\ \times\ 5\ 3\ -\ -3\ -8\ \times\ +\ 6\ 2\ 1\ 3\ -\ \times\ -\ +\ /$$ 6. **Prefix notation:** - Left subtree: $\times\ -3\ -5$ - Right subtree: - Inner left: $-\ 5\ 3$ - Inner right: $\times\ -3\ -8$ - Sum: $+\ -\ 5\ 3\ \times\ -3\ -8$ - Rightmost subtree: - $-\ 1\ 3$ - $\times\ 2\ -\ 1\ 3$ - $-\ 6\ \times\ 2\ -\ 1\ 3$ - Sum all right subtree parts: $$+\ +\ -\ 5\ 3\ \times\ -3\ -8\ -\ 6\ \times\ 2\ -\ 1\ 3$$ - Full prefix: $$/\ \times\ -3\ -5\ +\ +\ -\ 5\ 3\ \times\ -3\ -8\ -\ 6\ \times\ 2\ -\ 1\ 3$$ **Final answers:** - Lexicographic (infix): $$\frac{(-3 \times -5)}{(5 - 3 + (-3 \times -8) + (6 - 2 \times (-1 + 3)))}$$ - Postfix: $$-3\ -5\ \times\ 5\ 3\ -\ -3\ -8\ \times\ +\ 6\ 2\ 1\ 3\ -\ \times\ -\ +\ /$$ - Prefix: $$/\ \times\ -3\ -5\ +\ +\ -\ 5\ 3\ \times\ -3\ -8\ -\ 6\ \times\ 2\ -\ 1\ 3$$