Post on 26-Jun-2020
More Trees
Discrete Structures (CS 173) University of Illinois 1
Magritte
Rooted trees
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root
Terminology for rooted trees
Nodes: root, internal, leaf, level, tree height
Relations: parent/child/sibling, ancestor/descendant
3overhead
Another example
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1root
Another example
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1root
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1root
Decision trees
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Hierarchical data structure; HTML; XML
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Programs as trees
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Languages, parsing, parse trees
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More terminology
m-ary tree: each node can have at most m children
full: each node has either 0 children or 𝑚 children
complete: all leaves have the same height
balanced: all leaves are approximately the same height
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Tree terminology
Nodes: root, internal, leaf, level, tree height
Relations: parent/child/sibling, ancestor/descendant
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root
leaves
internalsubtree
level = 0
level = 1
level = 2
level = 3
More terminology for directed trees
m-ary tree: each node has at most m children
full: each node splits 0 or 𝑚 times
complete: all leaves have the same height
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Induction on rooted trees
A full m-ary tree with i internal nodes has mi + 1 nodes
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Induction proof on trees
Claim: In a binary tree of height ℎ, the number of nodes 𝑛 ≤ 2ℎ+1 − 1.
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Induction proof on trees
Claim: There are at most 𝑚ℎ leaves in an m-ary tree of height h
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Useful formulas
𝑘=0
𝑛
𝑟𝑘 =1 − 𝑟𝑛+1
1 − 𝑟
𝑘=𝑚
𝑛
𝑟𝑘 =𝑟𝑚 − 𝑟𝑛+1
1 − 𝑟
𝑘=𝑖
𝑛
𝑘 =𝑛(𝑛 + 1)
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𝑘=0
𝑛
2𝑘 =
𝑘=1
𝑛
2𝑘 =
𝑘=0
𝑛
2−2𝑘 =
𝑘=0
𝑛
2𝑘+2 =
𝑚𝑎 𝑏 = 𝑚𝑎𝑏2log2 𝑛 = 𝑛
log𝑎(𝑏) = log2(𝑏)/ log2 𝑎
𝑚𝑎+𝑏 = 𝑚𝑎𝑚𝑏
2log4(𝑛)+2 =
Tree induction proof
If 𝑇 is a binary tree with root 𝑟, then its rank is
(a) 0 if 𝑟 has no children
(b) 1 + 𝑞 if 𝑟 has two children, both with rank 𝑞
(c) otherwise, the maximum rank of any of the children
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Tree induction proof
If 𝑇 is a binary tree with root 𝑟, then its rank is
(a) 0 if 𝑟 has no children
(b) 1 + 𝑞 if 𝑟 has two children, both with rank 𝑞
(c) otherwise, the maximum rank of any of the children
Claim: A tree with rank 𝑞 has at least 2𝑞 leaves.
TRY TO DO THIS AS AN EXERCISE.
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