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

Grammars

• Syntactic Categories (non-terminals)– <number>– <digit>– <expr>

• Production Rules (replace syntactic category on the rhs by the lhs, “|” is or)– <expr> <expr> + <expr>– <expr> <number>– <number> <digit> <number>– <digit> 0|1|2|3|4|5|6|7|8|9

Derivation

• Repeatedly replace syntactic categories by the lhs of rules whose rhs is equal to the syntactic category

• <expr> <expr>+<expr>

<expr>+<expr>+<expr>

<number>+<expr>+<expr>

<number>+<number>+<expr>

<number>+<number>+<number>

Derivation (e.g. 2)

• <number> <digit><number> <digit><digit><number> <digit><digit><digit> <digit><digit>3 <digit>23 123• When there are no more syntactic categories, the

process stops and the resulting string is said to be derived from the initial syntactic category.

Languages

• The language, L(<S>), derivable from the syntactic category <S> using the grammar G is defined inductively.

• Initially L(<S>) is empty

• If <S> 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(<S>)

Language

• The number of strings of length n in the language L(<number>) is 10n.

• Proof is by induction.

Language

• <B> () | (<B>)

• L(<B>) = strings of n left parens followed by n right parens, for n >= 0.

Systematic Generation

• C statement example

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

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.

Number of Nodes of a Binary Trees

• Nnodes(T) = 0 if T is empty

• Nnodes(T.left) + Nnodes(T.right) + 1

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.

External Format for Binary Trees

• <bintree> [] [<value>,<bintree>,<bintree>]

• [], • [1,[],[]],• [2,[1,[],[]],[]], [2,[[],[1,[],[]]]• [3, [2,[1,[],[]],[]], []], [3, [2,[[],[1,[],[]]],[]] [3, [1,[],[]], [1,[],[]]], [3, [],[2,[1,[],[]],[]]], [3, [],[2,[[],[1,[],[]]]]

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

TTT kn

n

kkn 1

1

0

Binary Search Trees

• Binary Tree

• All elements in T->left are <= T->value

• All elements in T->right are >= T->value

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.

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

Example Parse Tree

<number> / \ <digit> <number> | / \ 1 <digit> <number> | | 2 <digit> | 3

Recursive Decent Parser

• Balanced parentheses

Ambiguous Grammars

<expr> <expr>

/ | \ / | \

<expr>+<expr> <expr>+<expr>

/ | \ / | \

<expr>+<expr> <expr>+<expr>