1. Trees
4/23/2017
Trees

2. 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
CAN
International
Europe
Asia
USA
Last Update: Oct 16, 2014
Trees

3. Tree Terminology For a node x:
Ancestors: x, parent, grandparent, great-grandparent, etc.
Descendants: x, children, grandchildren, great-grandchildren, etc.
Proper ancestors/descendants: exclude x itself.
Depth: number of proper ancestors of x.
For a tree:
Root: node without parent (A)
Internal node: node with at least one child (A, B, C, F)
External node (aka, leaf ): node without children (E, I, J, K, G, H, D)
Height: maximum node depth (3)
Subtree: tree consisting of a node and its descendants A B C D E F G H I J K
Last Update: Oct 16, 2014
Trees

4. A recursive view of TreesTk
Last Update: Oct 16, 2014
Trees

5. Tree ADT Query methods: We use positions to abstract nodes
Generic methods:
integer size()
boolean isEmpty()
Iterator iterator()
Iterable positions()
Accessor methods:
position root()
position parent(p)
Iterable children(p)
Integer numChildren(p)
Query methods:
boolean isInternal(p)
boolean isExternal(p)
boolean isRoot(p)
Additional update methods may be defined by data structures implementing the Tree ADT
Last Update: Oct 16, 2014
Trees

6. Java Tree interface
Last Update: Oct 16, 2014
Trees

7. Tree Traversals
A traversal method is a systematic way to explore the tree structure by visiting its nodes in a specific order.
Very useful computational tool with many applications.
Important tree traversals:
Preorder
Postorder
Inorder
Euler tour (generalizes the above three)
Level order (aka, Breadth First Search)  not shown in these slides. Generalized BFS on graphs will be discussed later in the course.
Last Update: Oct 16, 2014
Trees

8. Preorder Traversal Algorithm preOrder(v)
visit(v)
for each child w of v
preorder (w)
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
Make Money Fast!
1. Motivations
References
2. Methods
2.1 Stock Fraud
2.2 Ponzi Scheme
1.1 Greed
1.2 Avidity
2.3 Bank Robbery 1 2 3 5 4 6 7 8 9
Last Update: Oct 16, 2014
Trees

9. 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
EECS2011/ 8 3 7
todo.txt 1K
assignments/
programs/ 1 2 4 5 6
A1.doc 3K
A2.doc 2K
LinOpt.java 10K
Stocks.java 25K
Robot.java 20K
Last Update: Oct 16, 2014
Trees

10. Binary Trees Applications: A binary tree is a tree such that:
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
A leaf node has no children
Recursive definition of a binary tree T:
T consists of an external root node, or
T has internal root whose left and right subtrees are binary trees.
Applications:
arithmetic expressions
decision processes
searching A B C F G D E H I J
Last Update: Oct 16, 2014
Trees

11. A recursive view of Binary Trees
left subtree of
node
right subtree of
Last Update: Oct 16, 2014
Trees

12. 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
Last Update: Oct 16, 2014
Trees

13. 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?
How about coffee?
On expense account?
Starbucks
Chipotle
Gracie’s
Café Paragon
Yes No
Last Update: Oct 16, 2014
Trees

14. 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 = i + e = 2e - 1
h  i
h  (n - 1)/2
e  2h
h  log2 e
h  log2 (n + 1) - 1
Last Update: Oct 16, 2014
Trees

15. BinaryTree ADT
The BinaryTree ADT extends the Tree ADT, i.e., inherits all methods of Tree ADT
Additional methods:
position left(p)
position right(p)
position sibling(p)
boolean isInternal(p)
boolean isExternal(p)
The above position methods return null when there is no left, right, or sibling of p, respectively
Update methods may be defined by data structures implementing the BinaryTree ADT
Last Update: Oct 16, 2014
Trees

16. Inorder Traversal Algorithm inOrder(v)
if isInternal(v) then inOrder( left(v) )
visit(v)
if isInternal(v) then inOrder( right(v) )
In an inorder traversal a node is visited after its left subtree and before its right subtree
Application: draw a binary tree. Planar node coordinates:
x(v) = inorder rank of v
y(v) = depth of v 3 1 2 5 6 7 9 8 4
Last Update: Oct 16, 2014
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17. Print Arithmetic Expressions
Algorithm printExpr(v)
if isInternal(v) then print( “(” )
printExpr ( left(v) )
print( v.element () )
if isInternal(v) then printExpr ( right(v) )
print ( “)” )
Specialization of 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))
Last Update: Oct 16, 2014
Trees

18. Evaluate Arithmetic Expressions
Specialization of postorder:
recursive method returning the value of a subtree
when visiting an internal node, combine the values of its left & right subtrees
Algorithm evalExpr(v)
if isExternal (v) then
return v.element()
else
x  evalExpr(left(v))
y  evalExpr(right(v))
  v.element()
return x  y +  - 2 5 1 3
Last Update: Oct 16, 2014
Trees

19. Euler Tour Traversal Generic traversal of a binary tree
Includes as special cases: preorder, postorder and inorder
Walk around the tree and visit each node three times:
on the left (preorder)
from below (inorder)
on the right (postorder) +  - 2 5 1 3 L B R
Last Update: Oct 16, 2014
Trees

20. Template Method Pattern
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(); // local variable if tree.isExternal(p) { visitExternal(p, r); } else { visitLeft(p, r); r.leftResult = eulerTour(tree.left(p)); visitBelow(p, r); r.rightResult = eulerTour(tree.right(p)); visitRight(p, r); } return r.finalResult; } }
Generic algorithm that can be specialized by redefining certain steps
Implemented by means of an abstract Java class
Visit methods can be redefined by subclasses
Template method eulerTour
Recursively called on the left and right children
A local variable r of type Result with fields leftResult, rightResult and finalResult keeps track of the output of the recursive calls to eulerTour
Last Update: Oct 16, 2014
Trees

21. Specializations of EulerTour
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 ); }
// … the rest omitted … }
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)
Last Update: Oct 16, 2014
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22. Linked Structure for Trees B A D F C E
element
parent
children
child list
node
A node is represented by an object storing
Element
Parent node
list of children nodes
Node objects implement the Position ADT B D A C E F
Last Update: Oct 16, 2014
Trees

23. Linked 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 C E B D A C E
Last Update: Oct 16, 2014
Trees

24. Array-Based Representation of Binary Trees1 2 5 6 3 4 9 10 a h g f e d c b j
Nodes are stored in an array A a b d g h … 1 2 9 10
Node v is stored at A[rank(v)]
rank(root) = 0
rank(left(node)) = 2  rank(node) + 1
rank(right(node)) = 2  rank(node) + 2
rank(parent(node)) = (rank(node) -1)/2
Last Update: Oct 16, 2014
Trees

25. Comparison Linked Structure Array
Requires explicit representation of 3 links per position:
parent, left child, right child
Data structure grows as needed – no wasted space.
Parent and children are implicitly represented:
Lower memory requirements per position
Memory requirements determined by height of tree. If tree is sparse, this is highly inefficient.
Last Update: Oct 16, 2014
Trees

26. Summary The Tree ADT, tree terminologies, and Java interface
Tree Traversals
Preorder
Postorder
Inorder
Euler Tour
Level order (aka, breadth first search)
Binary trees: properties & some applications
Linked list & array based representations of trees.
Last Update: Oct 16, 2014
Trees

27. Last Update: Oct 16, 2014
Trees