CS210- Lecture 10
June 21, 2005
Announcements
 Assignment 3 has been posted
 Part 1 is due next Monday
 Part 2 is due next Tuesday
 Midterm is on June 30
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Section 3.1 - 3.4
Chapter 4
Section 5.1 - 5.3, Section 5.5
Part of Chapter 6 that I will be able to cover till this
Thursday (06/23).
 I will return Assignment 1 on Thursday (06/23)
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CS210-Summer 2005, Lecture 10
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Agenda
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Binary Trees
Properties of Binary Trees
Binary Tree Traversals
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Linear data Structures
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Here are some of the data structures
we have studied so far:
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Arrays
Stacks, Queues and deques
Singly linked list and doubly linked list
These all have the property that their
elements can be adequately displayed
in a straight line.
Binary trees are one of the simplest
nonlinear data structures.
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Parts of a binary tree
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A binary tree is composed of zero or more
nodes.
Each node can have at most two children.
Each node contains:
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A value (some sort of data item)
A reference or pointer to a left child (may be null)
A reference or pointer to a right child (may be
null)
A binary tree may be empty (contain no
nodes)
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Parts of a Binary tree
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If not empty, a binary tree has a root
node
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Every node in the binary tree is reachable
from the root node by a unique path.
A node with neither a left child nor a
right child is called a leaf (external
node).
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Picture of a Binary Tree
a
c
b
f
e
d
g
h
i
j
k
k
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Binary search in an array
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Look at array location mid = (low + high)/2
Searching for 5: mid = ( 0 + 6)/2 = 3
high = (mid -1)= 2
Using a binary
search tree
Low = (mid + 1) = 2
mid = (0 + 2)/2 = 1
mid = (2 + 2)/2 = 2
7
0
2
1
2
3
4
5
6
3
5
7
11 13 17
2
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The Binary Tree ADT
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The binary tree is a specialization of a
tree that supports these additional
accessor methods:
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Left(v): Return the left child of v.
Right(v): Return the right child of v.
hasLeft(v): Test whether v has a left child.
hasRight(v): Test whether v has a right
child.
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Binary Tree interface
public interface BinaryTree extends Tree{
public Position left(Position v) throws
InvalidPositionException,
BoundaryViolationException;
…
}
Assumptions on running times:
left(v), right(v), hasLeft(v), hasRight(v)
take O(1) time
children(v) take O(1) time because now there
can be at most two children.
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Properties of Binary Trees
Level
0
1
2
3
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Properties of Binary trees
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Let
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T - binary tree and let
n - number of nodes
ne – number of external nodes
ni – number of internal nodes
h – height of T
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h + 1 <= n <= 2h+1 -1
1 <= ne <= 2h
h <= ni <= 2h – 1
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log(n+1) – 1 <= h <= n -1
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Tree Traversals
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A binary tree is defined recursively: it consists
of a root, a left subtree and a right subtree
To traverse (or walk) the binary tree is to visit
each node in the binary tree exactly once.
Tree traversals are naturally recursive.
Since a binary tree has three parts, there are
six possible ways to traverse the binary tree:
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root, left, right : preorder (root, right, left)
left, root, right: inorder (right, root, left)
left, right, root: postorder (right, left, root)
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Traversals of a Binary tree
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Preorder Traversal
Algorithm binaryPreorder(T, v):
perform the “visit” action for node v
if T.hasLeft(v) then
binaryPreorder (T, T.left(v))
if T.hasRight(v) then
binaryPreorder(T, T.right(v))
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Traversals of a Binary tree
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Inorder Traversal
Algorithm inorder(T, v):
if T.hasLeft(v) then
inorder (T, T.left(v))
perform the “visit” action for node v
if T.hasRight(v) then
inorder(T, T.right(v))
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Traversals of a Binary tree
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Postorder Traversal
Algorithm postorder(T, v):
if T.hasLeft(v) then
postorder (T, T.left(v))
if T.hasRight(v) then
postorder(T, T.right(v))
perform the “visit” action for node v
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