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Data Structures and
Algorithms for Information
Processing
Lecture 5: Trees
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Binary Trees
 A binary tree has nodes, similar to
nodes in a linked list structure.
 Data of one sort or another may be
stored at each node.
 Each node is either a leaf, having no
children, or an internal node, with one
or two children
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Binary Trees
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Some Terminology
• Root is the unique node with no parent
• Leaves or terminals are nodes with no
children
• A subtree is a node together with all its
descendents
• Level of a node is number of nodes on
path from the node to root
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Example use of Trees
 Each internal node
labeled by an operator
 Each leaf labeled by a
variable or numeric value
Arithmetic
Expressions
A*(((B+C)*(D*E))+F)
Tree determines a
unique value
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Example Use of Trees
Representing Chinese Characters
• Characters
composed
hierarchically into
boxes
• Boxes can be
oriented left-right,
top-down, or
outside-inside
• Each leaf labeled
by one of 300-400
radicals
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Binary vs. Binary-Unary
Important distinction :
Sometimes binary trees are
defined to allow internal nodes
with one or two children.
There is a big difference...
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Counting Trees
The number of binary trees with
n internal nodes is
1  2n 
 
n 1 n 
There is no such formula if we
allow unary internal nodes!
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Exercise
Write efficient Java functions
int numBTrees(int n)
int numBUTrees(int n)
that return the number of binary
trees and binary/unary trees,
respectively.
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Properties of Trees
• There is exactly one path connecting
any two nodes
• Any two nodes have a least common
ancestor
• A tree with N internal nodes has N+1
external nodes
(easy to prove by induction)
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
The Best Trees
A binary tree is full if internal
nodes fill every level, except
possibly the last.
The height of a full binary tree
with N internal nodes is
h  log 2 N  1
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Complete Binary Trees
• A complete tree is one having
and 2 n leaves
n
levels
• Complete trees have a simple
implementation using arrays
(How?)
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Class for Binary Nodes
public class BTNode
{
private Object data;
private BTNode left;
private BTNode right;
...
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Class for Binary Nodes
public BTNode(Object obj,
BTNode l,
BTNode r)
{
data = obj; left = l; right= r;
}
public boolean isLeaf()
{
return (left == null) &&
(right == null);
}
...
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Copying Trees
public static BTNode
treeCopy(BTNode t)
{
if (t == null) return null;
else
{
BTNode leftCopy =
treeCopy(t.left);
BTNode rightCopy =
treeCopy(t.right);
return new BTNode(t.data,
leftCopy,
rightCopy);
}
}
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Tree Traversals
P
M
E
S
A
L
R
A
T
Preorder traversal
90-723: Data Structures
and Algorithms for
Information Processing
E
E
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Tree Traversals
P
M
L
E
S
A
R
A
T
Inorder traversal
90-723: Data Structures
and Algorithms for
Information Processing
E
E
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Tree Traversals
P
M
E
S
A
L
R
A
T
Postorder traversal
90-723: Data Structures
and Algorithms for
Information Processing
E
E
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Tree Traversals
P
M
E
S
A
L
R
A
T
Level order traversal
90-723: Data Structures
and Algorithms for
Information Processing
E
E
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Preorder Traversal
 Process the root
 Process the nodes in the left subtree
 Process the nodes in the right subtree
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Preorder Print
public void preorderPrint()
{
System.out.println(data);
if (left != null)
left.preorderPrint();
if (right != null)
right.preorderPrint();
}
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Inorder Traversal
 Profess the nodes in the left subtree
 Process the root
 Process the nodes in the right subtree
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.
Inorder Print
public void preorderPrint()
{
if (left != null)
left.preorderPrint();
System.out.println(data);
if (right != null)
right.preorderPrint();
}
90-723: Data Structures
and Algorithms for
Information Processing
Lecture 5: Trees
Copyright © 1999, Carnegie Mellon. All Rights Reserved.