03/14/13

More Trees

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

Magritte

Last class: trees and CFGs

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

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root

leaves

internal

parent, child

subtree

level = 0

level = 1

level = 2

level = 3

This lecture: more trees

• Recursion trees for illustrating computation in recursive functions

• Another tree proof

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Useful formulas

∑𝑘=0

𝑛

𝑟 𝑘=1−𝑟𝑛+1

1−𝑟 ∑𝑘=𝑚

𝑛

𝑟 𝑘= 𝑟𝑚−𝑟𝑛+1

1−𝑟 ∑𝑘=𝑖

𝑛

𝑘=𝑛(𝑛+1)2

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∑𝑘=0

𝑛

2𝑘=∑𝑘=1

𝑛

2𝑘=¿

(𝑚𝑎 )𝑏=𝑚𝑎𝑏

2 log2𝑛=𝑛log𝑎(𝑏)=log2(𝑏)/ log2 (𝑎)

𝑚𝑎+𝑏=𝑚𝑎𝑚𝑏

2 log4 (𝑛)+2=¿

Recursion trees

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cost of input

cost of subproblemsplit in 2

How many levels before base case?Sum of values in each level?How many leaf nodes?Total cost = sum of leaf costs + sum of internal costs:

Recursion trees

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Recursion trees A A

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Tree induction proofIf is a binary tree with root , then its rank is

(a) if has no children(b) if has two children, both with rank (c) otherwise, the maximum rank of any of the children

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Tree induction proofIf is a binary tree with root , then its rank is

(a) if has no children(b) if has two children, both with rank (c) otherwise, the maximum rank of any of the children

Claim: A tree with rank has at least leaves.

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Have a great break!

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