1. Trees Make Money Fast! Stock Fraud Ponzi Scheme Bank Robbery
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Trees
Make Money Fast!
Stock Fraud
Ponzi Scheme
Bank Robbery
© 2010 Goodrich, Tamassia
Trees 1 1

2. What is a Tree? Abstract model of a hierarchical structure
Trees
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What is a Tree?
Abstract model of a hierarchical structure
Nodes with a parent- child relation
Single root node
Applications:
Organization charts
File systems
Programming environments
Computers”R”Us
Sales
R&D
Manufacturing
Laptops
Desktops US
International
Europe
Asia
Canada
© 2010 Goodrich, Tamassia
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3. Tree Terminology subtree Root: node without parent (A).
Trees
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Tree Terminology
Root: node without parent (A).
Internal node: node with at least one child (A, B, C, F).
External node (leaf ): node without children (E, I, J, K, G, H, D).
Ancestors: parent, grandparent, grand-grandparent, etc.
Descendant: child, grandchild, grand-grandchild, etc.
Depth of node: maximum number of descendants.
Height of tree: depth of root (3).
Subtree: tree consisting of a node and its descendants (C, G, H). A B D C G H E F I J K
subtree
© 2010 Goodrich, Tamassia
Trees

4. Preorder Traversal Algorithm preOrder(v) visit(v)
Trees
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Preorder Traversal
A traversal visits the nodes of a tree in a systematic manner.
In a preorder traversal, a node is visited before its descendants.
Application: print a structured document.
Algorithm preOrder(v)
visit(v)
for each child w of v
preorder (w) 1
Make Money Fast! 2 5 9
1. Motivations
2. Methods
References 6 7 8 3 4
2.1 Stock Fraud
2.2 Ponzi Scheme
2.3 Bank Robbery
1.1 Greed
1.2 Avidity
© 2010 Goodrich, Tamassia
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5. Postorder Traversal Algorithm postOrder(v) for each child w of v
Trees
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Postorder Traversal
In a postorder traversal, a node is visited after its descendants.
Application: compute space used by files in a directory and its subdirectories.
Algorithm postOrder(v)
for each child w of v
postOrder (w)
visit(v) 9
cs16/ 8 3 7
todo.txt 1K
homeworks/
programs/ 1 2 4 5 6
h1c.doc 3K
h1nc.doc 2K
DDR.cpp 10K
Stocks.cpp 25K
Robot.cpp 20K
© 2010 Goodrich, Tamassia
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6. Binary Tree A binary tree is a tree with the following properties:
Trees
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Binary Tree
A binary tree is a tree with the following properties:
Each internal node has at most two children (exactly two for proper binary trees).
The children of a node are an ordered pair.
We call the children of an internal node left child and right child.
Alternative recursive definition: a binary tree is either
a tree consisting of a single node, or
a tree whose root has an ordered pair of children, each of which is a binary tree.
Applications:
arithmetic expressions
decision processes
searching A B C D E F G H I
© 2010 Goodrich, Tamassia
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7. Arithmetic Expression Tree
Trees
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Arithmetic Expression Tree
Binary tree associated with an arithmetic expression
internal nodes: operators
external nodes: operands
Example: arithmetic expression tree for the expression (2 (a  1)  (3 b))    2 a 1 3 b
© 2010 Goodrich, Tamassia
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8. Decision Tree Binary tree associated with a decision process
Trees
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Decision Tree
Binary tree associated with a decision process
internal nodes: questions with yes/no answer
external nodes: decisions
Example: dining decision
Want a fast meal?
Yes No
How about coffee?
On expense account?
Yes No
Yes No
Starbucks
Spike’s
Al Forno
Café Paragon
© 2010 Goodrich, Tamassia
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9. Properties of Proper Binary Trees
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Properties of Proper Binary Trees
Notation
n number of nodes
e number of external nodes
i number of internal nodes
h height
Properties:
e  i  1
n  2e  1
h i
h (n  1)2
e  2h
h  log2 e
h  log2 (n  1)  1
© 2010 Goodrich, Tamassia
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10. Inorder Traversal Algorithm inOrder(v) if  v.isExternal()
Trees
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Inorder Traversal
Each node is visited after its left subtree and before its right subtree.
Algorithm inOrder(v)
if  v.isExternal()
inOrder(v.left())
visit(v)
inOrder(v.right()) 6 2 8 1 4 7 9 3 5
© 2010 Goodrich, Tamassia
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11. Printing Arithmetic Expressions
Trees
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Printing Arithmetic Expressions
Algorithm printExpression(v)
if v.isExternal() print(“(’’)
inOrder(v.left())
print(v.element())
if v.isExternal()
inOrder(v.right())
print (“)’’)
Perform inorder traversal
print operand or operator when visiting node
print “(“ before traversing left subtree
print “)“ after traversing right subtree    2 a 1 3 b
((2 (a  1))  (3 b))
© 2010 Goodrich, Tamassia
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12. Linked Structure for Trees
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Linked Structure for Trees
A node is represented by an object storing
Element
Parent node
list of children nodes B  A  D F  B A D F C E C  E 
© 2010 Goodrich, Tamassia
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13. Linked Structure for Binary Trees
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Linked Structure for Binary Trees
A node is represented by an object storing
Element
Parent node
Left child node
Right child node  B   A D B A D     C E C E
© 2010 Goodrich, Tamassia
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14. Array Representation of Binary Trees
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Array Representation of Binary Trees
Nodes are stored in an array A 1 A A B D … G H … 2 3 1 2 3 10 11 B D
Node v is stored at A[rank(v)]
rank(root) = 1
if node is the left child of parent(node), rank(node) = 2  rank(parent(node))
if node is the right child of parent(node), rank(node) = 2  rank(parent(node))  1 4 5 6 7 E F C J 10 11 G H
© 2010 Goodrich, Tamassia
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15. Trees
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Binary Search Tree
A binary search tree is a binary tree storing keys (or key-value entries) at its internal nodes.
For nodes u, v, and w with u in the left subtree of v and w in the right subtree of v, key(u)  key(v)  key(w).
External nodes do not store items.
Inorder traversal visits the keys in increasing order. 6 9 2 4 1 8
© 2010 Goodrich, Tamassia
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16. Search To search for a key k, trace a downward path from the root.
Dictionaries
Trees
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Search
Algorithm TreeSearch(k, v)
if v.isExternal ()
return v
if k  v.key()
return TreeSearch(k, v.left())
else if k  v.key()
else { k  v.key() }
return TreeSearch(k, v.right())
To search for a key k, trace a downward path from the root.
The next node visited depends on the comparison of k with the key of the current node.
If a leaf is reached, the key is not in the tree.
Example: TreeSearch(4, root)  6 2  9 1 4  8
© 2010 Goodrich, Tamassia
Binary Search Trees 16 16

17. Insertion  Search for key k using TreeSearch.
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Insertion  6
Search for key k using TreeSearch.
Assume k is not already in the tree, and let w be the leaf reached by the search.
Insert k at node w and expand w into an internal node.
Example: insert 5.
Duplicate keys can be handled with non-strict inequality. 2 9  1 4 8  w 6 2 9 1 4 8 w 5
© 2010 Goodrich, Tamassia
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18. Deletion To delete k, search for key k. 
Trees
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Deletion 6
To delete k, search for key k.
If key k is in the tree, let v be the node storing k.
If node v has a leaf child w, remove v and w from the tree with removeExternal(w), which removes w and its parent.
Example: remove 4.  2 9  v 1 4 8 w 5 6 2 9 1 5 8
© 2010 Goodrich, Tamassia
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19. Trees
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Deletion (cont.) 1 v
If key k is stored at a node v with two internal children, find the internal node w that follows v in an inorder traversal.
Copy key(w) into node v.
Remove node w and its left child z (which must be a leaf) by means of operation removeExternal(z).
Example: remove 3. 3 2 8 6 9 w 5 z 1 v 5 2 8 6 9
© 2010 Goodrich, Tamassia
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20. Performance A binary tree with n keys takes O(n) space.
Trees
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Performance
A binary tree with n keys takes O(n) space.
The height h is O(n) in the worst case.
The height h is O(log n) in the average case.
Find, insert, and delete take O(h) time.
© 2010 Goodrich, Tamassia
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