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1
Introduction to Trees
Chapter 7
2
Introduction to Trees

• In the middle of nineteenth century, Gustav
  Kirchhoff studied on trees in mathematics.
• Several years later, Arthur Cayley used them to
  study the structure of algebraic formulas.
• In 1951, Grace Hoppers use of them to represent
  arithmetic expressions.
• Hoppers work bears a strong resemblance todays
  binary tree formats.

3
Introduction to Trees

• Trees are used in computer science
• To represent algebraic formulas,
• As an efficient method for searching large
  dynamic lists,
• For such diverse applications as artificial
  intelligence systems,
• And encoding algorithms.

4
Basic Concepts

• A tree consists of a finite set of elements,
  called nodes.
• A tree consists of a finite set of directed
  lines, called branches.
• These braches connect the nodes.
• The branch is directed towards the node, it is an
  indegree branch.
• The branch is directed away from the node, it is
  an outdegree branch.

5
Basic Concepts

• The sum of the indegree and outdegree branches
  equals the degree
• of the node.
• If the tree is not empty, then the first node is
  called the root.
• The indegree of the root is zero.
• All of the nodes in a tree (exception of root)
  must have an indegree
• of exactly one.

Figure 7-1
6
Basic Concepts
Parent, Outdegree gt 0
Internal Nodes, is not a root or a leaf.
Child, indegree gt 0
Leaf, Outdegree 0

• Ancestor,
• is any node in the path
• from the root node.
• Descendent is,
• all nodes in the path from given
• node to the leaf.

Siblings, with the same parent.
7
Basic Concepts
Height of tree max. Level of leaf 1
Figure 7-2
8
Basic Concepts

• A subtree is any connected structure below the
  root.
• A subtree can be divided into subtrees.

Figure 7-3
9
Root ( B ( C D ) E F ( G H I) )?

• algorithm ConvertToParent(val root ltnode
  pointergt, ref output ltstringgt)?
• Convert a general tree to parenthetical notation.
• Pre root is a pointer to a tree node.
• Post output contains parenthetical notation.
• Place root in output
• If (root is parent)?
• Place an open parenthesis in the output
• ConvertToParent(roots first child)?
• While (more siblings)?
• ConvertToParent(roots next child)?
• Place close parenthesis in the output
• Return
• End ConvertToParent

10
Binary Trees
A binary tree is a tree in which no node can have
more than two subtrees.
Figure 7-5
11
Binary Trees
Null tree
Symmetry is not a tree requirement!
Figure 7-6
12
Binary Trees

• Maximum height of tree for N nodes
• Hmax N
• The minimum height of the tree
• Hmin log2N 1
• If known the height of a tree
• Nmin H
• Nmax 2H-1

13
Binary Trees - Balance

• A complete tree has the maximum number of entries
  for its heigh. Nmax 2H-1
• The distance of a node from the root determines
  how efficiently it can be located.
• The balance factor show that the balance of
  the tree.
• B HL HR
• If B 0, 1 or -1 the tree is balanced.

Figure 7-7
14
Binary Tree Structure

• Each node in the structure must contain the data
  to be stored and two pointers, one to the left
  subtree and one to the right subtree.
• Node
• leftSubTree ltpointer to Nodegt
• data ltdataTypegt
• rightSubTree ltpointer to Nodegt
• End Node

15
Figure 7-25
16
Binary Tree Traversals

• A binary tree travelsal requires each node of
  the tree be processed once.
• In the depth-first traversal, all of the
  descendents of a child are processed before the
• next child.
• In the breadth-first traversal, each level is
  completely processed before the next level
• is started.

LRN
NLR
LNR
Three different depth-first traversal sequences.
Figure 7-8
17
Binary Tree Traversals
Preorder ? Inorder ? Postorder ?
Figure 7-9
18
Binary Tree Traversals - Preorder
Figure 7-10
19
Binary Tree Traversals - Preorder
Figure 7-11
Recursive algorithmic traversal of binary tree.
20
Binary Tree Traversals - Inorder
Figure 7-12
21
Binary Tree Traversals - Postorder
Figure 7-13
22
Binary Tree Breadth-First Traversals
Figure 7-14
23
Expression Trees

• An expression tree is a binary tree with these
  properties
• Each leaf is an operand.
• The root and internal nodes are operators.
• Subtrees are subexpressions with the root being
  an operator.

Figure 7-15
24
Infix Traversal Of An Expression Tree
Figure 7-16
25
Infix Traversal Of An Expression Tree

• algorithm infix (val tree lttree pointergt)?
• Print the infix expression for an expression
  tree.
• Pre tree is a pointer to an expression tree
• Post the infix expression has been printed
• 1. If (tree not empty)?
• 1. if (tree-gttoken is an operand)?
• 1. print (tree-gttoken)?
• 2 else
• 1. print (open parenthesis)?
• 2. infix(tree-gtleft)?
• 3. print(tree-gttoken)?
• 4. infix(tree-gtright)?
• 5. print(close parenthesis)?
• 2. Return
• end infix

26
Huffman Code

• ASCII 7 bits each character
• Some characters occur more often than others,
  like 'E'
• Every character uses the maximum number of bits
• Huffman, makes it more efficient
• Assign shorter codes to ones that occur often
• Longer codes for the ones that occur less often
• Typical frequencies

27
Huffman Code

• Organize character set into a row, ordered by
  frequency.
• Find two nodes with smallest combined weight,
  join them and form a third.
• Repeat until ALL are combined into a tree..

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Huffman...
29
Huffman
30
Huffman...

• Now we assign a code to each character
• Assign bit value for each branch
• 0 left branch,
• 1 right branch.
• A character's code is found by starting at root
  and following the branches.

31
Huffman

• Note that the letters that occur most
  often are represented with very few bits

32
General Trees

• A general tree is a tree which each node can have
  an unlimited outdegree.
• Binary trees are presented easier then general
  trees in programs.
• In general tree, there are two releationships
  that we can use
• Parent to child and,
• Sibling to sibling.
• Using these two relationships, we can represent
  any general tree as a binary tree.

33
Converting General Trees To Binary Trees
Figure 7-17
34
Insertion Into General Trees
FIFO insertion the nodes are inserted at the end
of the sibling list, (like insertion at the rear
of the queue).
Figure 7-18
35
Insertion Into General Trees
LIFO insertion places the new node at the
beginning of the sibling list, (like insertion
at the front of the stack).
Figure 7-19
36
Insertion Into General Trees
Key-sequence insertion places the new node in
key sequence among the sibling nodes.
Figure 7-20
37
Excercises

• Show the tree representation of the following
  parenthetical notation
• a ( b c ( e (f g) ) h )?

38
Excercises

• Find
• Root
• Leaves
• Internal nodes
• Ancestors of H
• Descendents of F
• Indegree of F
• Outdegree of B
• Level of G
• Heigh of node I

Figure 7-21
39
Excercises
What is the balance factor of the below tree?
Figure 7-22
40
Exercises
Find the infix, prefix and postfix expressions of
the below tree.
Figure 7-23
41
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42
HW-7

• Write a program to
• Create the following binary tree and
• Create a menu to select the printing of infix,
  prefix and postfix expressions of the tree.
• Print the tree selected expression type.

Load your HW-6 to FTP site until 04 May. 07 at
0900 am.