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
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Trees

2. Outline and Reading Tree ADT (§2.3.1)
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Outline and Reading
Tree ADT (§2.3.1)
Preorder and postorder traversals (§2.3.2)
BinaryTree ADT (§2.3.3)
Inorder traversal (§2.3.3)
Euler Tour traversal (§2.3.3)
Template method pattern
Data structures for trees (§2.3.4)
Java implementation (
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3. Trees
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What is a Tree
In computer science, a tree is an abstract model of a hierarchical structure
A tree consists of nodes with a parent-child relation
Applications:
Organization charts
File systems
Programming environments
Computers”R”Us
Sales
R&D
Manufacturing
Laptops
Desktops US
International
Europe
Asia
Canada
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4. Tree Terminology subtree Root: node without parent (A)
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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 (a.k.a. leaf ): node without children (E, I, J, K, G, H, D)
Ancestors of a node: parent, grandparent, grand-grandparent, etc.
Depth of a node: number of ancestors
Height of a tree: maximum depth of any node (3)
Descendant of a node: child, grandchild, grand-grandchild, etc.
Subtree: tree consisting of a node and its descendants A B D C G H E F I J K
subtree
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5. Tree ADT We use positions to abstract nodes Generic methods:
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Tree ADT
We use positions to abstract nodes
Generic methods:
integer size()
boolean isEmpty()
objectIterator elements()
positionIterator positions()
Accessor methods:
position root()
position parent(p)
positionIterator children(p)
Query methods:
boolean isInternal(p)
boolean isExternal(p)
boolean isRoot(p)
Update methods:
swapElements(p, q)
object replaceElement(p, o)
Additional update methods may be defined by data structures implementing the Tree ADT
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6. Preorder Traversal Algorithm preOrder(v) visit(v)
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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
Recursive traversal algorithm
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
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7. Postorder Traversal Algorithm postOrder(v) for each child w of v
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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.java 10K
Stocks.java 25K
Robot.java 20K
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8. Amortized Analysis of Tree Traversal
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Amortized Analysis of Tree Traversal
Time taken in preorder or postorder traversal of an n-node tree is proportional to the sum, taken over each node v in the tree, of the time needed for the recursive call for v.
The call for v costs $(cv + 1), where cv is the number of children of v
For the call for v, charge one cyber-dollar to v and charge one cyber-dollar to each child of v.
Each node (except the root) gets charged twice: once for its own call and once for its parent’s call.
Therefore, traversal time is O(n).
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9. Binary Trees Applications:
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Binary Trees
Applications:
arithmetic expressions
decision processes
Searching (IP address lookup)
A binary tree is a tree with the following properties:
Each internal node has two children
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 A B C D E F G H I
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10. Arithmetic Expression Tree
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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))
Inorder for expression, postorder for evaluation +  - 2 a 1 3 b
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11. Decision Tree Binary tree associated with a decision process
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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
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12. Properties of Binary Trees
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Properties of 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
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13. Trees
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BinaryTree ADT
The BinaryTree ADT extends the Tree ADT, i.e., it inherits all the methods of the Tree ADT
Additional methods:
position leftChild(p)
position rightChild(p)
position sibling(p)
Update methods may be defined by data structures implementing the BinaryTree ADT
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14. Inorder Traversal Algorithm inOrder(v) if isInternal (v)
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Inorder Traversal
In an inorder traversal a node is visited after its left subtree and before its right subtree
Application: draw a binary tree
x(v) = inorder rank of v
y(v) = depth of v
Algorithm inOrder(v)
if isInternal (v)
inOrder (leftChild (v))
visit(v)
inOrder (rightChild (v)) 6 2 8 1 4 7 9 3 5
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15. Print Arithmetic Expressions
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Print Arithmetic Expressions
Algorithm printExpression(v)
if isInternal (v) print(“(’’)
inOrder (leftChild (v))
print(v.element ())
if isInternal (v)
inOrder (rightChild (v))
print (“)’’)
Specialization of an 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))
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16. Evaluate Arithmetic Expressions
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Evaluate Arithmetic Expressions
Specialization of a postorder traversal
recursive method returning the value of a subtree
when visiting an internal node, combine the values of the subtrees
Algorithm evalExpr(v)
if isExternal (v)
return v.element ()
else
x  evalExpr(leftChild (v))
y  evalExpr(rightChild (v))
  operator stored at v
return x  y +  - 2 5 1 3
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17. Euler Tour Traversal +   2 - 3 2 5 1
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Euler Tour Traversal
Generic traversal of a binary tree
Includes special cases of the preorder, postorder and inorder traversals
Walk around the tree and visit each node three times:
on the left (preorder)
from below (inorder)
on the right (postorder) + L  R  B 2 - 3 2 5 1
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18. Template Method Pattern
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Template Method Pattern
Generic algorithm that can be specialized by redefining certain steps
Implemented by means of an abstract Java class
Visit methods that can be redefined by subclasses
Template method eulerTour
Recursively called on the left and right children
A Result object with fields leftResult, rightResult and finalResult keeps track of the output of the recursive calls to eulerTour
public abstract class EulerTour { protected BinaryTree tree; protected void visitExternal(Position p, Result r) { } protected void visitLeft(Position p, Result r) { } protected void visitBelow(Position p, Result r) { } protected void visitRight(Position p, Result r) { } protected Object eulerTour(Position p) { Result r = new Result(); if tree.isExternal(p) { visitExternal(p, r); } else { visitLeft(p, r); r.leftResult = eulerTour(tree.leftChild(p)); visitBelow(p, r); r.rightResult = eulerTour(tree.rightChild(p)); visitRight(p, r); return r.finalResult; } …
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19. Specializations of EulerTour
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Specializations of EulerTour
We show how to specialize class EulerTour to evaluate an arithmetic expression
Assumptions
External nodes store Integer objects
Internal nodes store Operator objects supporting method
operation (Integer, Integer)
public class EvaluateExpression extends EulerTour {
protected void visitExternal(Position p, Result r) { r.finalResult = (Integer) p.element(); }
protected void visitRight(Position p, Result r) { Operator op = (Operator) p.element(); r.finalResult = op.operation( (Integer) r.leftResult, (Integer) r.rightResult ); } … }
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20. Data Structure for Trees
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Data Structure for Trees
A node is represented by an object storing
Element
Parent node
Sequence of children nodes
Node objects implement the Position ADT  B   A D F B A D F   C E C E
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21. Data Structure for Binary Trees
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Data Structure for Binary Trees
A node is represented by an object storing
Element
Parent node
Left child node
Right child node
Node objects implement the Position ADT  B   A D B A D     C E C E
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22. Array-Based Representation of Binary Trees
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nodes are stored in an array 1 A H G F E D C B J … 2 3
let rank(node) be defined as follows:
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 10 11
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23. removeAboveExternal(w)
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Java Implementation
Tree interface
BinaryTree interface extending Tree
Classes implementing Tree and BinaryTree and providing
Constructors
Update methods
Print methods
Examples of updates for binary trees
expandExternal(v)
removeAboveExternal(w)
expandExternal(v) v A v A 
removeAboveExternal(w) A C B B w
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24. InspectableBinaryTree
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Trees in JDSL
JDSL is the Library of Data Structures in Java
Tree interfaces in JDSL
InspectableBinaryTree
InspectableTree
BinaryTree
Tree
Inspectable versions of the interfaces do not have update methods
Tree classes in JDSL
NodeBinaryTree
NodeTree
JDSL was developed at Brown’s Center for Geometric Computing
See the JDSL documentation and tutorials at
InspectableTree
Tree
InspectableBinaryTree
BinaryTree
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