03/12/13

03/12/13

Trees and CFGs

Discrete Structures (CS 173)Derek Hoiem, University of Illinois 1

Last class: recursive functions

base case recursive formula

• Process for finding closed form1. Unroll for several steps2. Write in terms of or 3. Substitute for value of that is base case4. Substitute base case value(s) and solve

• Verifying closed form with induction

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Today’s lecture: Trees and CFGs

• Trees– Examples of uses– Terminology– Induction on trees

• Context free grammars (CFGs)– What they are, how they work– Induction on CFG

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Tree: special form of graph with “root” and no cycles

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root

Tree terminologyNodes: root, internal, leaf, level, tree heightRelations: parent/child/sibling, ancestor/descendant

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Another example

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64

7

5

2

3

1root

Another example

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6

4

7

5

2

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1root

64

7

5

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1root

Trees for sorting

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><

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Decision trees

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Hierarchical data structure

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Trees for clustering• Goal: create a function that maps from to

such that nearby N-dimensional points are mapped to the same integer

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b>5

xx

x

x

x

x

x

xx

xx

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2 3

a

b

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a>6

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yes

no

no

yes

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More terminologym-ary tree: each node can split into m subtreesfull: each node splits or timescomplete: all leaves have the same heightBalanced: all leaves are approximately the same height

How many nodes in a full m-ary tree with internal nodes?

What are the lower/upper bounds on the number of nodes of an m-ary tree with height ?

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Induction proof on treesClaim: In a binary tree of height , the number of nodes .

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Context-free GrammarsA context-free grammar (CFG) is a set of rules that defines a set of possible parse trees.

A CFG specifies a set of rules, valid start symbols, and valid terminals.

Example: Start symbol: Terminals:

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CFG ExampleExample:

Which of these strings can be generated by the grammar above?

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Examples of parse trees

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Language

Fig: Johnson 2007

Examples of parse trees

17Figs: Zhu and Mumford 2007

Scene parse

Object Parse

Examples of parse trees

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Stochastic CFG for blackjack actions

Induction proof on CFG Claim: For any string generated by the grammar above, the number of ’s will be equal to the number of ’s plus the number of ’s (

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CFG example

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*

I like to eat apples and bananas

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Things to remember

• Trees are a special graph with root and no cycles, with many uses– Sorting, clustering, finding similar values– Decision tree: machine learning, modeling choices– Parse trees: representing hierarchical structures

• Context free grammars: generate parse trees

• Proofs on trees: split at root, use inductive hypothesis on subtrees headed by the root’s children

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Next class: more trees• Recursion trees and more proofs with trees

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