CFG Parse Trees

A Parse Tree, also known as a derivation tree, is a graphical representation of the derivation of a string in a context-free grammar (CFG). It visually illustrates how a string is generated from the grammar’s production rules, with each node representing a grammar symbol and its children representing the result of applying a production rule.

Structure of a Parse Tree

A parse tree has the following components:

  • Root: The root node represents the start symbol of the grammar.
  • Internal Nodes: These nodes represent non-terminal symbols of the grammar.
  • Leaf Nodes: The leaf nodes represent terminal symbols or the empty string (ε).
  • Edges: Edges connect nodes, showing the derivation of one symbol into others based on production rules.

The parse tree follows the hierarchical structure of the grammar, with each level corresponding to a step in the derivation.

Example of a Parse Tree

Consider the CFG for balanced parentheses:

</>
Copy
S → (S) | ε

For the string (()), the parse tree is as follows:

Include a diagram showing:

  • The root node S.
  • Two child nodes: ( and S, with S expanding into another (S).
  • The leaf nodes (, ), and the final empty string ε for the innermost S.

This shows the hierarchical application of production rules to generate the string (()).

Ambiguity in Parse Trees

A grammar is ambiguous if there exist multiple parse trees (or derivations) for the same string. Ambiguity can lead to confusion in parsing and is undesirable in programming language grammars.

Example: Consider the grammar:

</>
Copy
E → E + E | E * E | id

For the string id + id * id, there are two possible parse trees:

  • One where + has higher precedence than *.
  • Another where * has higher precedence than +.

Ambiguity is resolved by rewriting the grammar or applying precedence rules explicitly.

Applications of Parse Trees

Parse trees are extensively used in various areas:

  • Compilers: Parse trees help in syntax analysis, ensuring source code adheres to the language grammar.
  • Natural Language Processing: Parse trees analyze the grammatical structure of sentences in human languages.
  • Mathematical Expressions: Parse trees evaluate and represent arithmetic and logical expressions.

Key Takeaways

  • A parse tree represents the derivation of a string in a CFG in a hierarchical structure.
  • It is useful for syntax analysis in compilers and other language-processing tasks.
  • Ambiguity in parse trees indicates problems in the grammar that must be resolved.
  • Parse trees provide insight into the structure and meaning of strings generated by a CFG.