trees. outline and reading tree definitions and adt (§7.1) tree traversal algorithms for general...
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Trees
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)
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&DManufacturing
Laptops DesktopsUS International
Europe Asia Canada
subtee
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.
A
B DC
G HE F
I J K
• Subtree: tree consisting of a node and its descendants
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 DC
G HE F
I J K
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
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
C
E
B
DA
C E
F
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
C
E
B
DA
C E
F
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
C
E
B
DA
C E
F
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
DA
C E
F
B
A D F
C
E
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 Bibliography2. Object-Oriented Design
2.1 Goals, Principles,and Patterns
2.2 Inheritance andPolymorphism
1.1 Basic C++ Programming Elements
1.2 Expressions 2.3 Templates
1
2
3
5
46
7 8
9
Exercise: Preorder Traversal• In a preorder traversal, a node is
visited before its descendants• List the nodes of this tree in
preorder traversal order.
Algorithm preOrder(v)visit(v)for each child w of v
preOrder (w)
A
B DC
G HE F
I J K
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)
cs16/
homeworks/ todo.txt1Kprograms/
DDR.java10K
Stocks.java25K
h1c.doc3K
h1nc.doc2K
Robot.java20K
9
3
1
7
2 4 5 6
8
Exercise: Postorder Traversal
Algorithm postOrder(v)for each child w of v
postOrder(w)visit(v)
• In a postorder traversal, a node is visited after its descendants
• List the nodes of this tree in postorder traversal order.
A
B DC
G HE F
I J K
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
F GD E
H I
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
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?
How about coffee? On expense account?
Starbucks Antonio’s Veritas Chili’s
Yes No
Yes No Yes No
Properties of Binary Trees
• Notationn number of nodesl number of leavesi number of internal
nodesh height
• Properties:• l i 1• n 2l 1• h i• h (n 1)2• l 2h
• h log2 l
• h log2 (n 1) 1
Properties of Binary Trees
• Full (or proper) binary tree• All nodes have either 0 or 2 children
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
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
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
DA
C E
B
A D
C E
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)if isInternal(v)
inOrder(rightChild(v))
3
1
2
5
6
7 9
8
4
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.
Algorithm inOrder(v)if isInternal(v)
inOrder(leftChild(v))visit(v)if isInternal(v)
inOrder(rightChild(v))
A
B C
G HE F
I K
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
Print Arithmetic Expressions• Specialization of an inorder
traversal• print operand or operator when
visiting node• print “(“ before traversing left
subtree• print “)“ after traversing right
subtree
Algorithm printExpression(v)if isInternal(v)
print(“(’’)printExpression(leftChild(v))
print(v.element())if isInternal(v)
printExpression(rightChild(v))print (“)’’)
2
a 1
3 b((2 (a 1)) (3 b))
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 vreturn x y
2
a 1
3 b
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
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
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
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 2
LB
R
Euler Tour TraversalAlgorithm eulerTour(p) left-visit-action(p) if isInternal(p)
eulerTour(p.left())bottom-visit-action(p)if isInternal(p)
eulerTour(p.right())right-visit-action(p)
End Algorithm
2
5 1
3 2
LB
R
Print Arithmetic Expressions• 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 “)”
Algorithm printExpression(p)if isExternal(p) then
print value stored at pElse
print “(“printExpression(p.left())Print the operator stored at pprintExpression(p.right())
print “)”Endif
End Algorithm
2
a 1
3 b((2 (a 1)) (3 b))
Template Method Pattern• 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
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
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};