1. Trees

2. Outline and Reading Tree Definitions and ADT (§7.1)
Tree Traversal Algorithms for General Trees (preorder and postorder) (§7.2)
BinaryTrees (§7.3)
Data structures for trees (§7.1.4 and §7.3.4)
Traversals of Binary Trees (preorder, inorder, postorder) (§7.3.6)
Euler Tours (§7.3.7)

3. 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
Computers”R”Us
Sales
R&D
Manufacturing
Laptops
Desktops US
International
Europe
Asia
Canada

4. Tree Terminology Root: node without parent (A)
Internal node: node with at least one child (A, B, C, F)
Leaf (aka External node): node without children (E, I, J, K, G, H, D)
Ancestors of a node: parent, grandparent, great-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, great-grandchild, etc.
Subtree: tree consisting of a node and its descendants A B D C G H E F I J K
subtee

5. Exercise: Trees
Answer the following questions about the tree shown on the right:
What is the size of the tree (number of nodes)?
Classify each node of the tree as a root, leaf, or internal node
List the ancestors of nodes B, F, G, and A. Which are the parents?
List the descendents of nodes B, F, G, and A. Which are the children?
List the depths of nodes B, F, G, and A.
What is the height of the tree?
Draw the subtrees that are rooted at node F and at node K. A B D C G H E F I J K

6. 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 isLeaf (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

7. A Linked Structure for General 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 D A C E F   C E

8. A Linked Structure for General 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 D A C E F   C E

9. A Linked Structure for General 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 D A C E F   C E

10. A Linked Structure for General 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 D A C E F   C E

11. 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)
Data Structures & Algorithms
1. A C++ Primer
Bibliography
2. Object-Oriented Design
2.1 Goals, Principles,
and Patterns
2.2 Inheritance and
Polymorphism
1.1 Basic C++
Programming Elements
1.2 Expressions
2.3 Templates 1 2 3 5 4 6 7 8 9

12. Exercise: Preorder Traversal
In a preorder traversal, a node is visited before its descendants
List the nodes of this tree in preorder traversal order. A B D C G H E F I J K
Algorithm preOrder(v)
visit(v)
for each child w of v
preOrder (w)

13. 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

14. Exercise: Postorder TraversalB D C G H E F I J K
In a postorder traversal, a node is visited after its descendants
List the nodes of this tree in postorder traversal order.
Algorithm postOrder(v)
for each child w of v
postOrder(w)
visit(v)

15. Binary Tree 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
Applications:
arithmetic expressions
decision processes
searching A B C D E F G H I

16. Arithmetic Expression Tree
Binary tree associated with an arithmetic expression
internal nodes: operators
leaves: operands
Example: arithmetic expression tree for the expression (2  (a - 1) + (3  b)) +  - 2 a 1 3 b

17. Decision Tree Binary tree associated with a decision process
internal nodes: questions with yes/no answer
leaves: decisions
Example: dining decision
Want a fast meal?
Yes No
How about coffee?
On expense account?
Yes No
Yes No
Starbucks
Antonio’s
Veritas
Chili’s

18. Properties of Binary Trees
Notation
n number of nodes
l number of leaves
i number of internal nodes
h height
Properties:
l = i + 1
n = 2l - 1
h  i
h  (n - 1)/2
l  2h
h  log2 l
h  log2 (n + 1) - 1

19. Properties of Binary Trees
Full (or proper) binary tree
All nodes have either 0 or 2 children 

20. Properties of Binary Trees
Complete binary tree with height h
All 2h-1 nodes exist at height h-1
Nodes at level h fill up left to right 

21. 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

22. A 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 B A D     C E C E

23. 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

24. Exercise: Inorder Traversal
In an inorder traversal a node is visited after its left subtree and before its right subtree
List the nodes of this tree in inorder traversal order. A B C E F G H
Algorithm inOrder(v)
if isInternal(v)
inOrder(leftChild(v))
visit(v)
inOrder(rightChild(v)) I K

25. Exercise: Preorder & InOrder Traversal
Draw a (single) binary tree T, such that
Each internal node of T stores a single character
A preorder traversal of T yields EXAMFUN
An inorder traversal of T yields MAFXUEN

26. Print Arithmetic Expressions
Algorithm printExpression(v)
if isInternal(v) print(“(’’)
printExpression(leftChild(v))
print(v.element())
if isInternal(v)
printExpression(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))

27. 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 isLeaf(v)
return v.element()
else
x  evalExpr(leftChild(v))
y  evalExpr(rightChild(v))
  operator stored at v
return x  y +  - 2 a 1 3 b

28. Exercise: Arithmetic Expressions
Draw an expression tree that has
Four leaves, storing the values 2, 4, 8, and 8
3 internal nodes, storing operations +, -, *, / (operators can be used more than once, but each internal node stores only one)
The value of the root is 24

29. Exercise: Arithmetic Expressions
Draw an expression tree that has
Four leaves, storing the values 2, 4, 8, and 8
3 internal nodes, storing operations +, -, *, / (operators can be used more than once, but each internal node stores only one)
The value of the root is 24 + +  8 8 2 4

30. Exercise: Arithmetic Expressions
Draw an expression tree that has
Four leaves, storing the values 1, 3, 4, and 8
3 internal nodes, storing operations +, -, *, / (operators can be used more than once, but each internal node stores only one)
The value of the root is 24

31. Euler Tour Traversal Generic traversal of a binary tree
Includes as special cases 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) +  - 2 5 1 3 L B R

32. Euler Tour Traversal Algorithm eulerTour(p) left-visit-action(p)
if isInternal(p)
eulerTour(p.left())
bottom-visit-action(p)
eulerTour(p.right())
right-visit-action(p)
End Algorithm +  - 2 5 1 3 L B R

33. Print Arithmetic Expressions
Algorithm printExpression(p)
if isExternal(p) then print value stored at p
Else
print “(“
printExpression(p.left())
Print the operator stored at p
printExpression(p.right())
print “)”
Endif
End Algorithm
Specialization of an Euler Tour traversal
Left-visit-action: if node is internal, print “(“
Bottom-visit-action: print value or operator stored at node
Right-visit-action: if node is internal, print “)” +  - 2 a 1 3 b
((2  (a - 1)) + (3  b))

34. Template Method Pattern
class EulerTour { protected: BinaryTree* tree;
virtual void visitLeaf(Position p, Result r) { }
virtual void visitLeft(Position p, Result r) { } virtual void visitBelow(Position p, Result r) { } virtual void visitRight(Position p, Result r) { } int eulerTour(Position p) { Result r = initResult(); if (tree–>isLeaf(p)) {
visitLeaf(p, r);
return r.finalResult; } 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; } // … other details omitted
Generic algorithm that can be specialized by redefining certain steps
Implemented by means of an abstract C++ 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

35. Specializations of EulerTour
We show how to specialize class EulerTour to evaluate an arithmetic expression
Assumptions
External nodes support a function value(), which returns the value of this node.
Internal nodes provide a function operation(int, int), which returns the result of some binary operator on integers.
class EvaluateExpression : public EulerTour { protected: void visitLeaf(Position p, Result r) { r.finalResult = p.element().value(); }
void visitRight(Position p, Result r) { Operator op = p.element().operator(); r.finalResult = op.operation( r.leftResult, r.rightResult); }
// … other details omitted };