data structure and algorithm chapter 8 computer

COM267
Chapter 9: Trees
Data Structures Using C,
Second Edition
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Data Structures Using C, Second
Edition
Reema Thareja

 Introduction
 Types of Trees
 Creating a Binary Tree from a General Tree
 Traversing a Binary Tree
 Huffman’s Tree
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Data Structures Using C, Second Edition
Reema Thareja

Introduction
 A tree is recursively defined as a set of one or more
nodes where one node is designated as the root of
the tree and all the remaining nodes can be
partitioned into non-empty sets each of which is a
sub-tree of the root.
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 Node A is the root node, nodes B, C, and D are
children of the root node and form sub-trees of the
tree rooted at node A.
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Introduction
 Basic Terminology
 Root node The root node R is the topmost node in the tree. If
R = NULL, then it means the tree is empty.
 Sub-trees If the root node R is not NULL, then the trees T1, T2,
and T3 are called the sub-trees of R.
 Leaf node A node that has no children is called the leaf node
or the terminal node.
 Path A sequence of consecutive edges is called a path. For
example, in Figure, the path from the root node A to node I is
given as: A, D, and I.
 Ancestor node An ancestor of a node is any predecessor
node on the path from root to that node. The root node does
not have any ancestors. In the tree given in Figure, nodes A,
C, and G are the ancestors of node K.
 Descendant node A descendant node is any successor node
on any path from the node to a leaf node. Leaf nodes do not
have any descendants. In the tree given in Figure, nodes C, G,
J, and K are the descendants of node A.
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Introduction
 Basic Terminology
 Level number Every node in the tree is assigned a level
number in such a way that the root node is at level 0,
children of the root node are at level number 1. Thus,
every node is at one level higher than its parent. So, all
child nodes have a level number given by parent’s level
number + 1.
 Degree Degree of a node is equal to the number of
children that a node has. The degree of a leaf node is
zero.
 In-degree In-degree of a node is the number of edges
arriving at that node.
 Out-degree Out-degree of a node is the number of
edges leaving that node.
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Types of Trees
 Trees are of following 6 types:
 1. General trees
 2. Forests
 3. Binary trees
 4. Binary search trees
 5. Expression trees
 6. Tournament trees
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Types of Trees
 General trees
 General trees are data structures that store elements
hierarchically.
 The top node of a tree is the root node and each node,
except the root, has a parent.
 A node in a general tree (except the leaf nodes) may
have zero or more sub-trees.
 General trees which have 3 sub-trees per node are
called ternary trees.
 However, the number of sub-trees for any node may be
variable. For example, a node can have 1 sub-tree,
whereas some other node can have 3 sub-trees.
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Types of Trees
 General trees
 Although general trees can be represented as ADTs,
there is always a problem when another sub-tree is
added to a node that already has the maximum number
of sub-trees attached to it.
 Even the algorithms for searching, traversing, adding,
and deleting nodes become much more complex as
there are not just two possibilities for any node but
multiple possibilities.
 A general tree when converted to a binary tree may not
end up being well formed or full, but the advantages of
such a conversion enable the programmer to use the
algorithms for processes that are used for binary trees
with minor modifications.
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Types of Trees
 Forests
 A forest is a disjoint union of trees.
 A set of disjoint trees (or forests) is obtained by deleting
the root and the edges connecting the root node to
nodes at level 1.
 We have already seen that every node of a tree is the
root of some sub-tree.
 Therefore, all the sub-trees immediately below a node
form a forest.
 A forest can also be defined as an ordered set of zero or
more general trees.
 While a general tree must have a root, a forest on the
other hand may be empty because by definition it is a
set, and sets can be empty.
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Types of Trees
 Forests
 We can convert a forest into a tree by adding a single
node as the root node of the tree. For example, Figure a
shows a forest and Figure b shows the corresponding
tree.
 Similarly, we can convert a general tree into a forest by
deleting the root node of the tree.
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Types of Trees
 Binary Trees
 A binary tree is a data structure that is defined as a collection of elements
called nodes. In a binary tree, the topmost element is called the root
node, and each node has 0, 1, or at the most 2 children.
 A node that has zero children is called a leaf node or a terminal node.
Every node contains a data element, a left pointer which points to the left
child, and a right pointer which points to the right child.
 The root element is pointed by a 'root' pointer. If root = NULL, then it means
the tree is empty.
 Figure shows a binary tree. In the figure, R is the root node and the two
trees T1 and T2 are called the left and right sub-trees of R. T1 is said to be the
left successor of R. Likewise, T2 is called the right successor of R.
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Types of Trees
 Binary Trees
 Note that the left sub-tree of the root node consists of the nodes: 2, 4, 5,
8, and 9. Similarly, the right sub-tree of the root node consists of nodes:
3, 6, 7, 10, 11, and 12.
 In the tree, root node 1 has two successors: 2 and 3. Node 2 has two
successor nodes: 4 and 5.
 Node 4 has two successors: 8 and 9. Node 5 has no successor. Node 3
has two successor nodes: 6 and 7. Node 6 has two successors: 10 and
11. Finally, node 7 has only one successor: 12.
 A binary tree is recursive by definition as every node in the tree contains
a left sub-tree and a right sub-tree. Even the terminal nodes contain an
empty left sub-tree and an empty right sub-tree.
 Look at Figure, nodes 5, 8, 9, 10, 11, and 12 have no successors and
thus said to have empty sub-trees.
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Types of Trees
 Binary Trees (Terminology)
 Parent If N is any node in T that has left successor S1 and right
successor S2, then N is called the parent of S1 and S2.
Correspondingly, S1 and S2 are called the left child and the right
child of N. Every node other than the root node has a parent.
 Level number Every node in the binary tree is assigned a level
number (refer Figure). The root node is defined to be at level 0.
The left and the right child of the root node have a level number
1. Similarly, every node is at one level higher than its parents.
So all child nodes are defined to have level number as parent's
level number + 1.
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Types of Trees
 Degree of a node It is equal to the number of children that a
node has. The degree of a leaf node is zero. For example, in
the tree, degree of node 4 is 2, degree of node 5 is zero and
degree of node 7 is 1.
 Sibling All nodes that are at the same level and share the
same parent are called siblings (brothers). For example, nodes
2 and 3; nodes 4 and 5; nodes 6 and 7; nodes 8 and 9; and
nodes 10 and 11 are siblings.
 Leaf node A node that has no children is called a leaf node or a
terminal node. The leaf nodes in the tree are: 8, 9, 5, 10, 11,
and 12.
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Types of Trees
 Similar binary trees Two binary trees T and T’ are said to be
similar if both these trees have the same structure. Figure shows
two similar binary trees.
 Copies Two binary trees T and T’ are said to be copies if they have
similar structure and if they have same content at the
corresponding nodes. Figure shows that T’ is a copy of T.
 Edge It is the line connecting a node N to any of its successors. A
binary tree of n nodes has exactly n – 1 edges because every node
except the root node is connected to its parent via an edge.
 Path A sequence of consecutive edges. For example, in Figure,
the path from the root node to the node 8 is given as: 1, 2, 4, and
8.
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Types of Trees
 Depth The depth of a node N is given as the length of the path
from the root R to the node N. The depth of the root node is
zero.
 Height of a tree It is the total number of nodes on the path from
the root node to the deepest node in the tree. A tree with only a
root node has a height of 1.
 A binary tree of height h has at least h nodes and at most 2h – 1
nodes. This is because every level will have at least one node
and can have at most 2 nodes. So, if every level has two nodes
then a tree with height h will have at the most 2h– 1 nodes as at
level 0, there is only one element called the root. The height of
a binary tree with n nodes is at least log2(n+1) and at most n.
 In-degree/out-degree of a node It is the number of edges
arriving at a node. The root node is the only node that has an
in-degree equal to zero.
 Similarly, out-degree of a node is the number of edges leaving
that node.
 Binary trees are commonly used to implement binary search
trees, expression trees, tournament trees, and binary heaps.
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Types of Trees
 Complete Binary Trees
 A complete binary tree is a binary tree that satisfies two
properties. First, in a complete binary tree, every level, except
possibly the last, is completely filled. Second, all nodes appear
as far left as possible.
 In a complete binary tree Tn, there are exactly n nodes and level
r of T can have at most 2r nodes.
 Figure shows a complete binary tree.
 Note that in Figure, level 0 has 20 = 1 node, level 1 has 21 = 2
nodes, level 2 has 22 = 4 nodes, level 3 has 6 nodes which is
less than the maximum of 23 = 8 nodes.
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Types of Trees
 Complete Binary Trees
 In Figure, tree T13has exactly 13 nodes. They have been
purposely labelled from 1 to 13, so that it is easy for the reader
to find the parent node, the right child node, and the left child
node of the given node.
 The formula can be given as—if K is a parent node, then its left
child can be calculated as 2 × K and its right child can be
calculated as 2 × K + 1. For example, the children of the node 4
are 8 (2 × 4) and 9 (2 × 4 + 1). Similarly, the parent of the node K
can be calculated as | K/2 |.
 Given the node 4, its parent can be calculated as | 4/2 | = 2. The
height of a tree Tnhaving exactly n nodes is given as: Hn = | log2
(n + 1) |
 This means, if a tree T has 10,00,000 nodes, then its height is
21.
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Types of Trees
 Extended Binary Trees
 A binary tree T is said to be an extended binary tree (or a 2-tree)
if each node in the tree has either no child or exactly two
children. Figure shows how an ordinary binary tree is converted
into an extended binary tree.
 In an extended binary tree, nodes having two children are
called internal nodes and nodes having no children are called
external nodes. In Figure, the internal nodes are represented
using circles and the external nodes are represented using
squares.
 To convert a binary tree into an extended tree, every empty
sub-tree is replaced by a new node.
 The original nodes in the tree are the internal nodes, and the
new nodes added are called the external nodes.
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Types of Trees
 Representation of Binary Trees in the Memory
 In the computer’s memory, a binary tree can be maintained
either by using a linked representation or by using a
sequential representation.
 Linked representation of binary trees In the linked
representation of a binary tree, every node will have three parts:
the data element, a pointer to the left node, and a pointer to the
right node.
 So in C, the binary tree is built with a node type given below.
struct node {
struct node *left;
int data;
struct node *right;
};
 Every binary tree has a pointer ROOT, which points to the root
element (topmost element) of the tree. If ROOT = NULL, then
the tree is empty. The schematic diagram of the linked
representation of the binary tree is shown in Figure given on
next slide.
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Types of Trees
 In this Figure, the left position is used to point to the left child of
the node or to store the address of the left child of the node.
The middle position is used to store the data. Finally, the right
position is used to point to the right child of the node or to store
the address of the right child of the node.
 Empty sub-trees are represented using X (meaning NULL).
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Types of Trees
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Types of Trees
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Types of Trees
 Sequential representation of binary trees Sequential
representation of trees is done using single or one-dimensional
arrays.
 Though it is the simplest technique for memory representation, it is
inefficient as it requires a lot of memory space.
 A sequential binary tree follows the following rules:
 A one-dimensional array, called TREE, is used to store the elements of
tree.
 The root of the tree will be stored in the first location. That is, TREE[1]
will store the data of the root element.
 The children of a node stored in location K will be stored in locations (2
× K) and (2 × K+1).
 The maximum size of the array TREE is given as (2h–1), where h is the
height of the tree.
 An empty tree or sub-tree is specified using NULL. If TREE[1]= NULL,
then the tree is empty.
 Figure given on next slide shows a binary tree and its
corresponding sequential representation. The tree has 11 nodes
and its height is 4.
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Types of Trees
 Sequential representation of binary trees
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Types of Trees
 Binary Search Trees
 A binary search tree, also known as an ordered binary
tree, is a variant of binary tree in which the nodes are
arranged in an order.
 We will discuss the concept of binary search trees and
different operations performed on them in the next
chapter.
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Types of Trees
 Expression Trees
 Binary trees are widely used to store algebraic expressions.
For example,vconsider the algebraic expression given as:
Exp = (a – b) + (c * d)
 This expression can be represented using a binary tree as
shown in Figure.
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Types of Trees
 Expression Trees
 Given the binary tree, write down the expression that it
represents.
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Expression for the above binary tree is:
[{(a/b) + (c*d)} ^ {(f % g)/(h – i)}]

Types of Trees
 Tournament Trees
 We all know that in a tournament, say of chess, n number of
players participate. To declare the winner among all these
players, a couple of matches are played and usually three
rounds are played in the game.
 In every match of round 1, a number of matches are played in
which two players play the game against each other.
 The number of matches that will be played in round 1 will
depend on the number of players. For example, if there are 8
players participating in a chess tournament, then 4 matches
will be played in round 1. Every match of round 1 will be
played between two players.
 Then in round 2, the winners of round 1 will play against each
other. Similarly, in round 3, the winners of round 2 will play
against each other and the person who wins round 3 is
declared the winner.
 Tournament trees are used to represent this concept.
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Types of Trees
 Tournament Trees
 In a tournament tree (also called a selection tree), each external node
represents a player and each internal node represents the winner of the
match played between the players represented by its children nodes.
 These tournament trees are also called winner trees because they are
being used to record the winner at each level.
 There are 8 players in total whose names are represented using a, b, c, d,
e, f, g, and h. In round 1, a and b; c and d; e and f; and finally g and h play
against each other.
 In round 2, the winners of round 1, that is, a, d, e, and g play against each
other.
 In round 3, the winners of round 2, a and e play against each other.
 In the tree, the root node a specifies the winner.
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Creating a Binary Tree from a General Tree
 The rules for converting a general tree to a binary tree
are given below. Note that a general tree is converted
into a binary tree and not a binary search tree.
 Rule 1: Root of the binary tree = Root of the general tree
 Rule 2: Left child of a node = Leftmost child of the node
in the binary tree in the general tree
 Rule 3: Right child of a node in the binary tree = Right
sibling of the node in the general tree
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 Convert the given general tree into a binary tree.
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Traversing a Binary Tree
 Traversing a binary tree is the process of visiting each
node in the tree exactly once in a systematic way.
 Unlike linear data structures in which the elements
are traversed sequentially, tree is a nonlinear data
structure in which the elements can be traversed in
many different ways.
 There are different algorithms for tree traversals.
 These algorithms differ in the order in which the
nodes are visited.
 In this section, we will discuss these algorithms.
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 Pre-Order Traversal
 To traverse a non-empty binary tree in pre-order, the following operations
are performed recursively at each node.
 The algorithm works by:
 1. Visiting the root node,
 2. Traversing the left sub-tree, and finally
 3. Traversing the right sub-tree.
 Consider the tree given in Figure. The pre-order traversal of the tree is
given as A, B, C.
 Root node first, the left sub-tree next, and then the right sub-tree. Pre-
order traversal is also called as depth-first traversal. In this algorithm, the
left sub-tree is always traversed before the right sub-tree.
 The word ‘pre’ in the pre-order specifies that the root node is accessed
prior to any other nodes in the left and right sub-trees. Pre-order
algorithm is also known as the NLR traversal algorithm(Node-Left-Right).
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 Pre-Order Traversal
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 In-Order Traversal
 To traverse a non-empty binary tree in in-order, the following operations are
performed recursively at each node.
 1. Traversing the left sub-tree,
 2. Visiting the root node, and finally
 3. Traversing the right sub-tree.
 Consider the tree given in Figure. The in-order traversal of the tree is given as B, A,
and C.
 Left sub-tree first, the root node next, and then the right sub-tree. In-order traversal
is also called as symmetric traversal.
 In this algorithm, the left sub-tree is always traversed before the root node and the
right sub-tree.
 The word ‘in’ in the in-order specifies that the root node is accessed in between the
left and the right sub-trees. In-order algorithm is also known as the LNR traversal
algorithm (Left-Node-Right).
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 In-Order Traversal
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 Post-Order Traversal
 To traverse a non-empty binary tree in post-order, the following
operations are performed recursively at each node.
 1. Traversing the left sub-tree,
 2. Traversing the right sub-tree, and finally
 3. Visiting the root node.
 Consider the tree given in Figure. The post-order traversal of the tree is
given as B, C, and A. Left sub-tree first, the right sub-tree next, and finally
the root node.
 In this algorithm, the left sub-tree is always traversed before the right
sub-tree and the root node. The word ‘post’ in the post-order specifies
that the root node is accessed after the left and the right sub-trees.
 Post-order algorithm is also known as the LRN traversal algorithm (Left-
Right-Node).
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 Post-Order Traversal
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 Level-Order Traversal
 In level-order traversal, all the nodes at a level are
accessed before going to the next level.
 This algorithm is also called as the breadth-first traversal
algorithm. Consider the trees given in Figure and note
the level order of these trees.
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 Constructing a Binary Tree from Traversal Results
 We can construct a binary tree if we are given at least two traversal results. The
first traversal must be the in-order traversal and the second can be either pre-order
or post-order traversal.
 The in-order traversal result will be used to determine the left and the right child
nodes, and the pre-order/post-order can be used to determine the root node. For
example, consider the traversal results given below:
In–order Traversal: D B E A F C G Pre–order Traversal: A B D E C F G
 Here, we have the in-order traversal sequence and pre-order traversal sequence.
Follow the steps given below to construct the tree:
 Step 1 Use the pre-order sequence to determine the root node of the tree. The first
element would be the root node.
 Step 2 Elements on the left side of the root node in the in-order traversal sequence
form the left sub-tree of the root node. Similarly, elements on the right side of the
root node in the in-order traversal sequence form the right sub-tree of the root
node.
 Step 3 Recursively select each element from pre-order traversal sequence and
create its left and right sub-trees from the in-order traversal sequence.
 Look at Figure which constructs the tree from its traversal results.
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Huffman’s Tree
 Huffman coding is an entropy encoding algorithm developed by
David A. Huffman that is widely used as a lossless data
compression technique.
 The Huffman coding algorithm uses a variable length code table
to encode a source character where the variable-length code
table is derived on the basis of the estimated probability of
occurrence of the source character.
 The key idea behind Huffman algorithm is that it encodes the
most common characters using shorter strings of bits than those
used for less common source characters.
 The algorithm works by creating a binary tree of nodes that are
stored in an array.
 A node can be either a leaf node or an internal node.
 Initially, all the nodes in the tree are at the leaf level and store the
source character and its frequency of occurrence (also known as
weight).
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Huffman’s Tree
 While the internal node is used to store the weight and contains
links to its child nodes, the external node contains the actual
character.
 Conventionally, a '0' represents following the left child and a '1'
represents following the right child.
 A finished tree that has n leaf nodes will have n – 1 internal
nodes.
 The running time of the algorithm depends on the length of the
paths in the tree.
 So, before going into further details of Huffman coding, let us first
learn how to calculate the length of the paths in the tree.
 The external path length of a binary tree is defined as the sum of
all path lengths summed over each path from the root to an
external node. The internal path length is also defined in the
same manner.
 The internal path length of a binary tree is defined as the sum of
all path lengths summed over each path from the root to an
internal node.
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Huffman’s Tree
 The internal path length, LI = 0 + 1 + 2 + 1 + 2 + 3 + 3 = 12
 The external path length, LE = 2 + 3 + 3 + 2 + 4 + 4 + 4 + 4 = 26
 Note that, LI + 2 * n = 12 + 2 * 7 = 12 + 14 = 26 = LE
 Thus, LI + 2n = LE, where n is the number of internal nodes.
 Now if the tree has n external nodes and each external node is
assigned a weight, then the weighted path length P is defined as
the sum of the weighted path lengths.
 Therefore, P = W1L1 + W2L2 + …. + WnLn where Wi and Li are the
weight and path length of an external node Ni.
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Huffman’s Tree 47
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Huffman’s Tree
Technique
 Given n nodes and their weights, the Huffman algorithm is used to find a
tree with a minimum weighted path length.
 The process essentially begins by creating a new node whose children are
the two nodes with the smallest weight, such that the new node’s weight is
equal to the sum of the children’s weight.
 That is, the two nodes are merged into one node. This process is repeated
until the tree has only one node. Such a tree with only one node is known as
the Huffman tree.
 The Huffman algorithm can be implemented using a priority queue in which
all the nodes are placed in such a way that the node with the lowest weight
is given the highest priority.
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Huffman’s Tree 49
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Huffman’s Tree 50
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Huffman’s Tree 51
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Huffman’s Tree
Data Coding
 When we want to code our data (character) using bits, then we use r bits to
code 2r characters.
 For example, if r=1, then two characters can be coded. If these two
characters are A and B, then A can be coded as 0 and B can be coded as 1
and vice versa.
 Look at Tables 9.1 and 9.2 which show the range of characters that can be
coded by using r=2 and r=3.
 Now, if we have to code the data string ABBBBBBAAAACDEFGGGGH, then
the corresponding code would be:
000001001001001001001000000000000010011100101110110110110111
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Huffman’s Tree
Data Coding
 This coding scheme has a fixed-length code because
every character is being coded using the same number of
bits.
 Although this technique of coding is simple, coding the
data can be made more efficient by using a variable-length
code.
 You might have observed that when we write a text in
English, all the characters are not used frequently. For
example, characters like a, e, i, and r are used more
frequently than w, x, y, z and so on.
 So, the basic idea is to assign a shorter code to the
frequently occurring characters and a longer to less
frequently occurring characters.
 Variable-length coding is preferred over fixed-length coding
because it requires lesser number of bits to encode the
same data.
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Huffman’s Tree
Data Coding
 For variable-length encoding, we first build a Huffman tree. First, arrange all
the characters in a priority queue in which the character with the lowest
frequency of occurrence has the highest priority.
 Then, create a Huffman tree as explained in the previous section. Figure
9.24 shows a Huffman tree that is used for encoding the data set.
 In the Huffman tree, circles contain the cumulative weights of their child
nodes.
 Every left branch is coded with 0 and every right branch is coded with 1. So,
the characters A, E, R, W, X, Y, and Z are coded as shown in Table 9.3.
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data structure and algorithm chapter 8 computer

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