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03/12/13

Trees and CFGs. 03/12/13. Discrete Structures (CS 173) Derek Hoiem, University of Illinois. Last class: recursive functions.  base case  recursive formula Process for finding closed form Unroll for several steps Write in terms of or Substitute for value of that is base case

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03/12/13

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  1. Trees and CFGs 03/12/13 Discrete Structures (CS 173) Derek Hoiem, University of Illinois

  2. Last class: recursive functions  base case  recursive formula • Process for finding closed form • Unroll for several steps • Write in terms of or • Substitute for value of that is base case • Substitute base case value(s) and solve • Verifying closed form with induction

  3. 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

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

  5. Tree terminology Nodes: root, internal, leaf, level, tree height Relations: parent/child/sibling, ancestor/descendant overhead

  6. Another example 7 4 6 root 1 5 2 3

  7. Another example root 1 7 4 6 2 root 1 5 2 3 4 6 3 5 7

  8. Trees for sorting < > < >

  9. Decision trees

  10. Hierarchical data structure

  11. Trees for clustering • Goal: create a function that maps from to such that nearby N-dimensional points are mapped to the same integer b>5 1 x no yes x x x a>6 5 1 x x no yes x x x x x 2 3 b 3 2 a 6

  12. More terminology m-ary tree: each node can split into m subtrees full: each node splits or times complete: all leaves have the same height Balanced: 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 ? overhead

  13. Induction proof on trees Claim: In a binary tree of height , the number of nodes . overhead

  14. Context-free Grammars A 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: overhead

  15. CFG Example Example: Which of these strings can be generated by the grammar above? overhead

  16. Examples of parse trees Language Fig: Johnson 2007

  17. Examples of parse trees Scene parse Object Parse Figs: Zhu and Mumford 2007

  18. Examples of parse trees Stochastic CFG for blackjack actions

  19. 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 ( overhead

  20. CFG example * I like to eat apples and bananas overhead

  21. 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

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

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