Question

Draw an arithmetic-expression tree that has four external nodes, storing the numbers \(1,5,6,\) and 7 (with each number stored in a distinct external node, but not necessarily in this order), and has three internal nodes, each storing an operator from the set \(\\{+,-, \times, /\\},\) so that the value of the root is \(21 .\) The operators may return and act on fractions, and an operator may be used more than once.

Short answer

The tree has 7 - 1 and 6 / 2 as internal nodes, combining to (7 - 1) * (6 / 2) = 21.

Step-by-step solution

- Understand the Problem

The goal is to create an arithmetic-expression tree with four external nodes (leaf nodes) storing the numbers 1, 5, 6, and 7, and three internal nodes (non-leaf nodes) storing the operators +, -, *, or /. The resultant value of the root should be 21.

- Determine Potential Operators

Consider the operators and how they could be applied to the numbers to achieve the target value of 21 in the root node. Operations must be planned to combine the given numbers using arithmetic operators.

- Formulate Partial Expressions

Try different combinations of the numbers with the operators to form partial expressions. For example: (7 - 1) * 3 = 18, or (5 * 6) / 2 = 15. Aim to achieve some intermediary values close to the target 21.

- Combine Partial Expressions

Combine the partial expressions formed in the previous step to see if they can result in the final value of 21. For example: (7 - 1) * (6 / 2) = 21, as (7 - 1) = 6 and (6 / 2) = 3, and 6 * 3 = 21.

- Draw the Tree

Draw the arithmetic-expression tree based on the combination that worked. The root node will contain the final operator, and branch out to the sub-expressions created in the previous step. For instance:- Root: * - Left Child: (7 - 1) - Left Grandchild: 7 - Right Grandchild: 1 - Right Child: (6 / 2) - Left Grandchild: 6 - Right Grandchild: 2

Key concepts

binary tree

A binary tree is a powerful data structure used in computing, where each node has at most two children. This structure is fundamental for representing arithmetic expressions efficiently. Binary trees allow us to visualize complex expressions in a hierarchical manner.
Here is how it works:

• Each node in the tree can be either an internal node or an external (or leaf) node.
• Internal nodes hold operators (like +, -, *, /).
• Leaf nodes store the operands (like numbers).

A binary tree for the expression \( (7 - 1) * (6 / 2) \) will look like this:
1. The root node (internal) holds the operator '*'.
2. The root has two children – these are sub-expressions.
3. The left child hosts another internal node with operator '-', leading to leaf nodes holding '7' and '1'.
4. The right child has an internal node with operator '/', leading to leaf nodes holding '6' and '2'.
By breaking down complex expressions into binary trees, we can visually and systematically solve arithmetic problems.

arithmetic operators

Arithmetic operators are the backbone of mathematical expressions. They define the operations performed on the numbers. In our context, the primary arithmetic operators are:

• Addition (+)
• Subtraction (-)
• Multiplication (*)
• Division (/)

These operators can act on integers and fractions, influencing the order and outcome of operations within an expression tree. For instance:

• The subtraction operation '7 - 1' gives us '6'.
• The division operation '6 / 2' results in '3'.

The order in which these operations are performed is crucial. This introduces us to the concept of operator precedence, where some operations are performed before others, unless specified otherwise through parentheses.

mathematical expressions

Mathematical expressions are combinations of numbers and arithmetic operators arranged in a meaningful way. They can represent anything from simple calculations to complex equations. Here's a breakdown:

• Operands: The numbers in the expression (e.g., 1, 5, 6, 7).
• Operators: The symbols that define the operations (e.g., +, -, *, /).

For example, the expression \( (7 - 1) * (6 / 2) \) can be deconstructed into smaller parts, solving each in sequence.
Each part of the expression needs to be handled following proper mathematical rules to ensure the correct solution. By using an arithmetic expression tree, we break down these expressions into understandable segments.

step-by-step problem solving

Step-by-step problem solving involves breaking down complex problems into simpler, manageable parts. For our arithmetic-expression tree problem, follow these steps:
1. **Understand the Problem**: Clearly identify the objective – in our case, constructing an expression tree with given numbers and operators to achieve the value of 21.
2. **Determine Potential Operators**: Evaluate how different operators can be applied to the given numbers to get close to the desired result.
3. **Formulate Partial Expressions**: Test various combinations of numbers and operators to find intermediate values. This step may involve some trial and error.
4. **Combine Partial Expressions**: Integrate the useful partial expressions into a coherent solution that satisfies the problem's requirements.
5. **Draw the Tree**: Finally, translate the combined expression into a binary tree structure, ensuring all numbers and operators are correctly positioned.
By following these steps methodically, even the most challenging problems can be solved effectively and logically.