Parse trees and derivations in Compiler Design - Step-by-Step Execution

Concept Flow - Parse trees and derivations

Start with Grammar

↓

Input String

↓

Apply Production Rules

↓

Build Parse Tree Step-by-Step

↓

Derivation Sequence

↓

Final Parse Tree and Derivation

The process starts with a grammar and an input string, applies production rules step-by-step to build a parse tree, showing the derivation sequence until the final parse tree is complete.

Execution Sample

Compiler Design

Grammar:
S -> a S b | ε
Input: a a b b
Derivation:
S => a S b => a a S b b => a a b b

Shows how the input string 'a a b b' is derived from the start symbol S using production rules, building the parse tree.

Analysis Table

Step	Current String	Production Applied	Resulting String	Parse Tree Node Added
1	S	S -> a S b	a S b	Root S with children 'a', S, 'b'
2	a S b	S -> a S b	a a S b b	Expand S node to 'a', S, 'b'
3	a a S b b	S -> ε	a a b b	Replace S with ε (empty)
4	a a b b	None	a a b b	Parse tree complete

💡 No more non-terminals to expand; input string fully derived.

State Tracker

Variable	Start	After 1	After 2	After 3	Final
Current String	S	a S b	a a S b b	a a b b	a a b b
Parse Tree Nodes	Root S	S with 'a', S, 'b'	Expanded S again	S replaced by ε	Complete tree

Key Insights - 2 Insights

Why do we replace S with ε in step 3?

Because the production S -> ε means S can be empty, which ends the expansion. This is shown in execution_table row 3 where the current string changes from 'a a S b b' to 'a a b b'.

How does the parse tree grow with each production?

Each production adds nodes to the parse tree representing the applied rule. For example, in step 1 and 2, the S node expands to 'a', S, 'b', adding branches as shown in execution_table rows 1 and 2.

Visual Quiz - 3 Questions

Look at the execution_table, what is the Current String after step 2?

A a S b

B a a S b b

C a a b b

D S

💡 Hint

Check the 'Current String' column in row 2 of the execution_table.

At which step is the production S -> ε applied?

A Step 3

B Step 2

C Step 1

D Step 4

💡 Hint

Look at the 'Production Applied' column to find where S -> ε is used.

If we did not use S -> ε, what would happen to the parse tree?

A It would be the same

B It would be smaller

C It would keep expanding infinitely

D It would be incomplete

💡 Hint

Refer to variable_tracker showing how S expands and the role of ε in stopping expansions.

Concept Snapshot

Parse trees show how input strings are generated from grammar rules.
Derivations are sequences of rule applications.
Start from start symbol, apply productions stepwise.
Each step expands non-terminals in the string.
Parse tree nodes represent grammar symbols and expansions.
ε means empty string, used to end expansions.

Full Transcript

Parse trees and derivations explain how a string is formed from grammar rules. Starting with the start symbol, we apply production rules step-by-step. Each step replaces a non-terminal with the right side of a rule, building the parse tree. The derivation sequence shows these replacements as strings. When a non-terminal is replaced by ε, it means it produces an empty string, ending that branch. The parse tree visually represents this process, with nodes for symbols and branches for expansions. This helps understand how compilers analyze code structure.

Practice

1. What does a parse tree represent in compiler design?

easy

A. The list of tokens from lexical analysis

B. The sequence of machine instructions generated

C. The memory layout of variables

D. The hierarchical structure of a sentence based on grammar rules

Solution

1. Step 1: Understand the role of parse trees

   Parse trees visually show how a sentence is broken down into parts according to grammar rules.

2. Step 2: Differentiate from other compiler components

   Unlike tokens or machine instructions, parse trees focus on syntax structure, not code or memory.

3. Final Answer:

   The hierarchical structure of a sentence based on grammar rules -> Option D

4. Quick Check:

   Parse tree = hierarchical structure [OK]

Hint: Parse tree = sentence structure, not code or memory [OK]

Common Mistakes:

• Confusing parse tree with tokens list
• Thinking parse tree shows machine code
• Mixing parse tree with memory layout

2. Which of the following is the correct way to represent a derivation step in grammar?

easy

A. S → aSb

B. S = a + Sb

C. S := aSb

D. S <- aSb

Solution

1. Step 1: Identify standard grammar notation

   In grammar, derivations use the arrow symbol → to show how a symbol expands.

2. Step 2: Check each option's correctness

   S → aSb uses → correctly; others use assignment or incorrect symbols.

3. Final Answer:

   S → aSb -> Option A

4. Quick Check:

   Derivation uses → symbol [OK]

Hint: Look for arrow → in grammar rules [OK]

Common Mistakes:

• Using assignment symbols instead of arrow
• Confusing programming assignment with grammar rules
• Using incorrect arrow symbols

3. Given the grammar rules:
S → aSb | ab
What is the leftmost derivation of the string aabb?

medium

A. S → aSb → aaSbb → aabb

B. S → ab → aabb

C. S → aSb → ab

D. S → aSb → aab

Solution

1. Step 1: Apply leftmost derivation rules

   Start with S, replace leftmost S using S → aSb to get aSb.

2. Step 2: Apply rule again to inner S

   Replace inner S with aSb again to get aaSbb, then replace S with ab to get aabb.

3. Final Answer:

   S → aSb → aaSbb → aabb -> Option A

4. Quick Check:

   Leftmost derivation = S → aSb → aaSbb → aabb [OK]

Hint: Replace leftmost non-terminal first [OK]

Common Mistakes:

• Replacing rightmost non-terminal first
• Skipping intermediate derivation steps
• Confusing production rules

4. Identify the error in this derivation sequence for grammar S → aSb | ab to produce aabb:
S → aSb → ab → aabb

medium

A. Used wrong production rule for S

B. String aabb cannot be derived from this grammar

C. Incorrectly replaced inner S with ab too early

D. No error, derivation is correct

Solution

1. Step 1: Analyze the derivation steps

   The derivation replaces S with aSb, then immediately replaces S with ab but in the wrong position.

2. Step 2: Check correctness of replacement order

   The inner S should be replaced after expanding the outer S fully; here it skips a step causing incorrect string formation.

3. Final Answer:

   Incorrectly replaced inner S with ab too early -> Option C

4. Quick Check:

   Early replacement error = Incorrectly replaced inner S with ab too early [OK]

Hint: Check order of non-terminal replacements carefully [OK]

Common Mistakes:

• Replacing non-terminals out of order
• Assuming all strings are derivable
• Ignoring grammar rules sequence

5. Consider the grammar:
E → E + T | T
T → T * F | F
F → (E) | id
Which parse tree correctly represents the expression id + id * id?

hard

A. Parse tree where '*' is root, left child '+' with two 'id' leaves, right child 'id'

B. Parse tree where '+' is root, left child 'id', right child '*' with two 'id' leaves

C. Parse tree with '+' as root and both children as 'id'

D. Parse tree with '*' as root and both children as 'id'

Solution

1. Step 1: Understand operator precedence in grammar

   Multiplication (*) has higher precedence than addition (+), so '*' groups tighter.

2. Step 2: Build parse tree accordingly

   The root is '+', left child is 'id', right child is '*', which has two 'id' children.

3. Final Answer:

   Parse tree where '+' is root, left child 'id', right child '*' with two 'id' leaves -> Option B

4. Quick Check:

   Operator precedence respected = Parse tree where '+' is root, left child 'id', right child '*' with two 'id' leaves [OK]

Hint: Remember * binds tighter than + in parse trees [OK]

Common Mistakes:

• Ignoring operator precedence
• Swapping '+' and '*' as root
• Misplacing leaves in parse tree