Question

Draw the binary tree representation of the following arithmetic expression: \("(((5+2) *(2-1)) /((2+9)+((7-2)-1)) * 8)\).

Short answer

The root is '*' with left child '/' and right child '8'. The '/' operator connects '*', '5+2', '2-1', '2+9', and '7-2-1'.

Step-by-step solution

Understand the Expression

The expression to represent as a binary tree is \( (((5+2) * (2-1)) / ((2+9) + ((7-2)-1))) * 8 \). Each sub-expression will be a node in the tree, with operators as parent nodes and operands as child nodes.

Identify the Main Operator

Break down the main structure of the expression. The main operator is the multiplication (*) of two parts: \( (((5+2) * (2-1)) / ((2+9) + ((7-2)-1))) \) and \( 8 \).

Split the Expression at the Main Operator

Create the root node of the tree with '*' and split the expression into left and right sub-expressions. The left sub-expression is \( (((5+2) * (2-1)) / ((2+9) + ((7-2)-1))) \), and the right is \( 8 \).

Break Down Left Sub-Expression

For the left sub-expression \( (((5+2) * (2-1)) / ((2+9) + ((7-2)-1))) \), the main operator is '/'. Create a new node with '/' as a child of '*'. Divide further into two parts: \( (5+2) * (2-1) \) and \( (2+9) + ((7-2)-1) \).

Break Down Multiplication in Left Sub-Expression

The sub-expression \( (5+2) * (2-1) \) has '*' as the main operator. Create a new node '*' and divide it into '5+2' and '2-1'.

Break Down Addition and Subtraction in Left Sub-Expression

The sub-expression \( 5+2 \) has '+' as the operator. Create a new node '+' and have '5' and '2' as child nodes. Similarly, '2-1' has '-' as the operator. Create a new node '-' and have '2' and '1' as child nodes.

Break Down Right Sub-Expression of the Main Operator

For the right sub-expression \( (2+9) + ((7-2)-1) \), the main operator is '+'. Create a new node '+' and divide into '2+9' and '((7-2)-1)'.

Break Down Further Sub-Expressions

The sub-expression \( 2+9 \) has '+' as the operator. Create a node '+' and have '2' and '9' as children. For '((7-2)-1)', the main operator is '-'. Create a new node '-' and divide '7-2' as one part and '1' as another. '7-2' has '-' as an operator, with '7' and '2' as children.

Assemble the Tree

Put all parts together into a binary tree format with the respective operators as parent nodes and operands as child nodes, building up from the leaves to the root.

Key concepts

arithmetic expressions

An arithmetic expression combines numbers and operation symbols (+, -, *, /) to represent a mathematical computation. These expressions follow a specific order of operations known as BODMAS/BIDMAS rules: Brackets, Orders (i.e., powers and square roots, etc.), Division and Multiplication (left to right), Addition and Subtraction (left to right). In this exercise, our main expression is (((5+2) * (2-1)) / ((2+9) + ((7-2)-1))) * 8. Each arithmetic operation, such as addition, subtraction, multiplication, or division, forms the basis of the nodes in our binary tree.

tree traversal

Tree traversal is a common technique used to explore all nodes in a tree data structure. There are three primary methods of traversal - Pre-order, In-order, and Post-order.Pre-order traversal visits nodes in the following sequence: root, left, then right. This method is helpful for creating a prefix expression.In-order traversal processes nodes in the order left, root, then right. It reflects the original arithmetic expression with parentheses removed.Post-order traversal explores nodes by visiting the left, right, and then the root. This technique helps create postfix expressions. By understanding these traversal methods, we can interpret the binary tree’s structure appropriately.

binary tree construction

Binary tree construction for arithmetic expressions involves turning each part of the expression into tree nodes. Operators serve as internal parent nodes, while operands are leaf nodes. Steps for building a binary tree from an expression are:

• Identify the main operator of the whole expression. It's the operator reached last following BODMAS rules.
• Recursively break down the expression into sub-expressions, each forming smaller trees.
• Continue splitting until reaching individual operands, making them leaf nodes of the tree.

By organizing the expression this way, you create a tree structure that visually represents the computation.

parsing expressions

Parsing expressions is the process of analyzing a string of symbols to determine its grammatical structure. For an arithmetic expression, parsing helps in understanding the hierarchy and order of operations. Steps include:

• Identify and separate operators and operands.
• Recognize and handle parentheses to prioritize sub-expressions.
• Determine the main operator of the expression to split it into sub-parts.

Effective parsing transforms the linear text of an expression into a hierarchical tree presentation, making complex calculations easier to manage and understand.