Master the Derivation Tree: Left & Right Derivation Trees Explained

Master the Derivation Tree: Left & Right Derivation Trees Explained

Table of Contents

1. Introduction
2. Understanding Derivation Trees
   1. Definition of a Derivation Tree
   2. Example Grammar
   3. Forming the Derivation Tree
      1. Root Vertex
      2. Non-Terminal Symbol Vertices
      3. Terminal Symbol Leaves
      4. Epsilon Leaves
3. Left Derivation Tree vs Right Derivation Tree
   1. Left Derivation Tree
   2. Right Derivation Tree
   3. Example
4. Conclusion

Introduction

In this article, we will Delve into the concept of derivation trees, also known as parse trees. Derivation trees are rooted, ordered, tree-like structures that graphically represent the semantic information of strings derived from a Context-free grammar. By understanding derivation trees, we can gain insights into how strings are generated from a given grammar. In this article, we will explore derivation trees in Detail, covering their formation, components, and the important distinction between left and right derivation trees.

Understanding Derivation Trees

Definition of a Derivation Tree

A derivation tree, also known as a parse tree, is an ordered, rooted tree that visually represents how strings can be generated from a context-free grammar. It serves as a graphical representation of the process of deriving strings from a grammar. The tree consists of vertices and leaves, with each vertex representing a non-terminal symbol (variable) and each leaf representing a terminal symbol or an epsilon (empty) symbol.

Example Grammar

To better understand derivation trees, let's consider an example grammar. The grammar is as follows:

G = (VT, P, S)

Where:

• VT refers to the variables or non-terminal symbols
• P refers to the set of production rules
• S refers to the start symbol

For our example, the grammar is:

• VT: B, a
• P:
  • S -> 0B
  • B -> 0aa
  • a -> 1aa
  • a -> ε
• S: S

In this grammar, we have two variables, B and a, and two terminal symbols, 0 and 1. The production rules dictate how the variables can be replaced to generate strings.

Forming the Derivation Tree

To form a derivation tree for a given grammar, we follow certain guidelines for labeling the vertices and leaves:

1. The root vertex should be labeled by the start symbol, which is S in our example.
2. Non-terminal symbol vertices should be labeled by the corresponding variables (B, a).
3. Terminal symbol leaves should be labeled by the terminal symbols (0, 1).
4. Epsilon leaves represent empty symbols and should be labeled by the epsilon symbol (ε).

Using the example grammar, let's see how the derivation tree is formed:

[Insert Derivation Tree Image]

In this derivation tree, the root vertex is labeled with the start symbol S. The other vertices (nodes) are labeled with the non-terminal symbols B and a. The leaves are labeled with the terminal symbols 0 and 1, as well as the epsilon symbol (ε).

Left Derivation Tree vs Right Derivation Tree

There are two types of derivation trees: left derivation trees and right derivation trees. The distinction between them lies in the order in which productions are applied to variables.

Left Derivation Tree

A left derivation tree is obtained by applying productions to the leftmost variable in each step. This means that in each derivation step, we choose the leftmost variable and apply the corresponding production. Left derivation trees are useful in understanding how strings are generated when the leftmost derivations are followed.

Right Derivation Tree

A right derivation tree is obtained by applying productions to the rightmost variable in each step. In contrast to left derivation trees, right derivation trees focus on the rightmost derivations. In each step, we choose the rightmost variable and apply the corresponding production.

Example

Let's consider the same example grammar and the STRING 'aaBa':

[Insert Example Derivation Trees Images]

By constructing both left and right derivation trees, we can observe the difference in the order of applying productions. Left derivation tree follows the leftmost derivation while right derivation tree follows the rightmost derivation.

Conclusion

Derivation trees provide a visual representation of the process of generating strings from a context-free grammar. Understanding derivation trees and the distinction between left and right derivation trees can enhance our understanding of how strings are formed. By following the guidelines for constructing derivation trees, we can analyze and interpret the structure and semantics of strings derived from grammars with ease.

In the next section, we will address frequently asked questions related to derivation trees and provide detailed answers to further clarify the concepts discussed in this article.

FAQ

Q: What is a derivation tree?

A: A derivation tree, also known as a parse tree, is an ordered, rooted tree that visually represents how strings can be generated from a context-free grammar.

Q: How do you label the vertices and leaves in a derivation tree?

A: The root vertex is labeled with the start symbol, while non-terminal symbol vertices are labeled with the corresponding variables. Terminal symbol leaves are labeled with the terminal symbols, and epsilon leaves are labeled with the epsilon symbol (ε).

Q: What is the difference between left derivation trees and right derivation trees?

A: Left derivation trees Apply productions to the leftmost variable in each step, while right derivation trees apply productions to the rightmost variable in each step.

Q: How do derivation trees help in understanding string generation?

A: Derivation trees provide a graphical representation of the process of deriving strings from a grammar. By analyzing the structure and components of the derivation tree, we can gain insights into how strings are formed.