Recursive Data Structures and Grammars

1
Recursive Data Structures and Grammars

• Themes
• Recursive Description of Data Structures
• Grammars and Parsing
• Recursive Definitions of Properties of Data
  Structures
• Recursive Algorithms for Manipulating and
  Traversing Data Structures
• Examples
• Lists
• Trees
• Expressions and Expression Trees

2
Grammars

• Syntactic Categories (non-terminals)
• ltnumbergt
• ltdigitgt
• ltexprgt
• Production Rules (replace syntactic category on
  the rhs by the lhs, is or)
• ltexprgt ? ltexprgt ltexprgt
• ltexprgt ? ltnumbergt
• ltnumbergt ? ltdigitgt ltnumbergt
• ltdigitgt ? 0123456789

3
Derivation

• Repeatedly replace syntactic categories by the
  lhs of rules whose rhs is equal to the syntactic
  category
• ltexprgt ? ltexprgtltexprgt
• ? ltexprgtltexprgtltexprgt
• ? ltnumbergtltexprgtltexprgt
• ? ltnumbergtltnumbergtltexprgt
• ? ltnumbergtltnumbergtltnumbergt

4
Derivation (e.g. 2)

• ltnumbergt ? ltdigitgtltnumbergt
• ? ltdigitgtltdigitgtltnumbergt
• ? ltdigitgtltdigitgtltdigitgt
• ? ltdigitgtltdigitgt3
• ? ltdigitgt23
• ? 123
• When there are no more syntactic categories, the
  process stops and the resulting string is said to
  be derived from the initial syntactic category.

5
Languages

• The language, L(ltSgt), derivable from the
  syntactic category ltSgt using the grammar G is
  defined inductively.
• Initially L(ltSgt) is empty
• If ltSgt ? X1 ? ? ? Xn is a production in G and si
  Xi is a terminal or si ? L(Xi), then the
  concatenation s1s2 sn is in L(ltSgt)

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Language

• The number of strings of length n in the language
  L(ltnumbergt) is 10n.
• Proof is by induction.

7
Language

• ltBgt ? () (ltBgt)
• L(ltBgt) strings of n left parens followed by n
  right parens, for n gt 0.

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Systematic Generation

• C statement example

9
Binary Trees

• A binary tree is
• empty
• consists of a node with 3 elements
• value
• left, which is a tree
• right, which is a tree

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Height of Binary Trees

• Height(T) -1 if T is empty
• max(Height(T.left),Height(T.right)) 1
• Alternative Max over all nodes of the level of
  the node.

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Number of Nodes of a Binary Trees

• Nnodes(T) 0 if T is empty
• Nnodes(T.left) Nnodes(T.right) 1

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Internal Path Length

• IPL(T) 0 if T is empty
• IPL(T) IPL(T.left) IPL(T.right) Nnodes(T)-1
• Alternative Sum over all nodes of the level of
  the node.

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External Format for Binary Trees

• ltbintreegt ?
• ? ltvaluegt,ltbintreegt,ltbintreegt
  
• ,
• 1,,,
• 2,1,,,, 2,,1,,
• 3, 2,1,,,, , 3,
  2,,1,,,
• 3, 1,,, 1,,,
• 3, ,2,1,,,, 3,
  ,2,,1,,

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Recurrence for the Number of Binary Trees

• Let Tn be the number of binary trees with n
  nodes.
• T0 1, T1 1, T2 2, T3 5

15
Binary Search Trees

• Binary Tree
• All elements in T-gtleft are lt T-gtvalue
• All elements in T-gtright are gt T-gtvalue

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Inorder traversal

• Recursively visit nodes in T.left
• visit root
• Recursively visit nodes in T.right
• An in order traversal of a BST lists the elements
  in sorted order. Proof by induction.

17
Parse Tree

• A derivation is conveniently stored in a tree,
  where internal nodes correspond to syntactic
  categories and the children of a node correspond
  to the element of the rhs in the rule that was
  applied

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Example Parse Tree

• ltnumbergt
• / \
• ltdigitgt ltnumbergt
• / \
• 1 ltdigitgt ltnumbergt
• 2 ltdigitgt
• 3

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Recursive Decent Parser

• Balanced parentheses

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Ambiguous Grammars

• ltexprgt ltexprgt
• / \ /
  \
• ltexprgtltexprgt ltexprgtltexprgt
• / \
  / \
• ltexprgtltexprgt ltexprgtltexprgt