Objectives. In this session, you will learn to:
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1 TRS
2 Objectives n this session, you will learn to: Store data in a tree istinguish types of inary tree Traverse a inary Tree norder PreOrder PostOrder onstruct a inary Tree onstruct an expression inary tree
3 Storing ata in a Tree onsider a scenario where you are required to represent the directory structure of your operating system. The directory structure contains various folders and files. folder may further contain any number of sub folders and files. n such a case, it is not possible to represent the structure linearly because all the items have a hierarchical relationship among themselves. n such a case, it would be good if you have a data structure that enables you to store your data in a nonlinear fashion. irectory structure:
4 efining Trees tree is a nonlinear data structure that represent a hierarchical relationship among the various data elements. Trees are used in applications in which the relation between data elements needs to be represented in a hierarchy. K J L M
5 efining Trees (ontd.) ach element in a tree is referred to as a node. The topmost node in a tree is called root. root K J L M node
6 efining Trees (ontd.) ach node in a tree can further have subtrees below its hierarchy. root K J L M node
7 Tree Terminology Let us discuss various terms that are most frequently used with trees. Leaf node: t refers to a node with no children. Nodes,,,,, J, L, and M are leaf nodes. K J L M
8 Tree Terminology (ontd.) Subtree: portion of a tree, which can be viewed as a separate tree in itself is called a subtree. subtree can also contain just one node called the root node. hildren of a node: The roots of the subtrees of a node are called the children of the node. Tree with root, containing nodes,,, and is a subtree of node.,,, and are children of node. is the parent of these nodes. K J L M
9 Tree Terminology (ontd.) dge: link from the parent to a child node is referred to as an edge. egree of a node: t refers to the number of subtrees of a node in a tree. egree of node is 1 egree of node is 2 egree of node is 3 egree of node is 4 dge K J L M
10 Tree Terminology (ontd.) Siblings/rothers: t refers to the children of the same node. Nodes,, and are siblings of each other. Nodes,,, and are siblings of each other. K J L M
11 Tree Terminology (ontd.) nternal node: t refers to any node between the root and a leaf node. Level of a node: t refers to the distance (in number of nodes) of a node from the root. Root always lies at level 0. s you move down the tree, the level increases by one. Level 0 Nodes,,, and K are internal nodes. Level 1 K J L Level 2 M Level 3
12 Tree Terminology (ontd.) epth of a tree: Refers to the total number of levels in the tree. The depth of the following tree is 4. Level 0 Level 1 K J L Level 2 M Level 3
13 Just a minute onsider the following tree and answer the questions that follow: a. What is the depth of the tree? b. Which nodes are children of node? c. Which node is the parent of node? d. What is the level of node? e. Which nodes are the siblings of node? f. Which nodes are the siblings of node? g. Which nodes are leaf nodes? root nswer: a. 4 b. and c. d. 2 e. does not have any siblings f. The only sibling of is g.,,, and
14 efining inary Trees inary tree is a specific type of tree in which each node can have at most two children namely left child and right child. There are various types of binary trees: Strictly binary tree ull binary tree omplete binary tree Strictly binary tree: binary tree in which every node, except for the leaf nodes, has non-empty left and right children.
15 efining inary Trees (ontd.) ull binary tree: binary tree of depth d that contains exactly 2d 1 nodes. epth = 3 Total number of nodes = 23 1 = 7
16 efining inary Trees (ontd.) omplete binary tree: binary tree with n nodes and depth d whose nodes correspond to the nodes numbered from 0 to n 1 in the full binary tree of depth k ull inary Tree omplete inary Tree ncomplete inary Tree
17 Representing a inary Tree rray representation of binary trees: ll the nodes are represented as the elements of an array inary Tree [0] [1] [2] [3] [4] [5] [6] f there are n nodes in a binary tree, then for any node with index i, where 0 < i < n 1: Parent of i is at (i 1)/2. Left child of i is at 2i + 1: f 2i + 1 > n 1, then the node does not have a left child. Right child of i is at 2i + 2: rray Representation f 2i + 2 > n 1, then the node does have a right child.
18 Representing a inary Tree (ontd.) Linked representation of a binary tree: t uses a linked list to implement a binary tree. ach node in the linked representation holds the following information: ata Reference to the left child Reference to the right child f a node does not have a left child or a right child or both, the respective left or right child fields of that node point to NULL. ata Node
19 Representing a inary Tree (ontd.) root root inary Tree Linked Representation 80
20 inary Tree Node Struct node { nt info; Struct node *left,*right; }
21 OPRTONS ON TRS Traversing a inary Tree 1)TRVRSN You can implement various operations on a binary tree. common operation on a binary tree is traversal. Traversal refers to the process of visiting all the nodes of a binary tree once. There are three ways for traversing a binary tree: norder traversal Preorder traversal Postorder traversal
22 NORR TRVRSL Steps for traversing a tree in inorder sequence are as follows: 1. Traverse the left subtree 2. Visit root 3. Traverse the right subtree Let us consider an example.
23 norder Traversal (ontd.) is is not not NULL. NULL. The left subtree of node to traverse the left Therefore, move to node. subtree of. root
24 norder Traversal (ontd.) The left subtree of node is NULL. Therefore, visit node. root
25 norder Traversal (ontd.) Right subtree of is not NULL Therefore, move to the right subtree of node root
26 norder Traversal (ontd.) Left subtree of is empty. Therefore, visit node. root
27 norder Traversal (ontd.) Right subtree of is empty. Therefore, move to node. root
28 norder Traversal (ontd.) The left subtree of has been visited. Therefore, visit node. root
29 norder Traversal (ontd.) Right subtree of is not empty. Therefore, move to the right subtree of. root
30 norder Traversal (ontd.) Left subtree of is empty. Therefore, visit node. root
31 norder Traversal (ontd.) Right subtree of is empty. Therefore, move to node. root
32 norder Traversal (ontd.) Left subtree of has been visited. Therefore, visit node. root
33 norder Traversal (ontd.) Right subtree of is not empty. Therefore, move to the right subtree of. root
34 norder Traversal (ontd.) Left subtree of is not empty. Therefore, move to the left subtree of. root
35 norder Traversal (ontd.) Left subtree of is empty. Therefore, visit node. root
36 norder Traversal (ontd.) Right subtree of is empty. Therefore, move to node. root
37 norder Traversal (ontd.) The left subtree of node has been visited. Therefore, visit node. root
38 norder Traversal (ontd.) Right subtree of is not empty. Therefore, move to the right subtree of node. root
39 norder Traversal (ontd.) Left subtree of is not empty. Therefore, move to the left subtree of node. root
40 norder Traversal (ontd.) Left subtree of is empty. Therefore, visit. root
41 norder Traversal (ontd.) Right subtree of is empty. Therefore, move to node. root
42 norder Traversal (ontd.) Visit node. root
43 norder Traversal (ontd.) Right subtree of is empty. root Traversal complete
44 LORTM LORTM NORRTRVRS() { 1. set top=0, stack[top]=null, ptr = root 2. Repeat while ptr!=null 2.1 set top=top set stack[top]=ptr 2.3 set ptr=ptr->left 3. Set ptr=stack[top], top=top-1 4. Repeat while ptr!=null 4.1 print ptr->info 4.2 if ptr->right!=null then 4.2.1set ptr=ptr->right goto step Set ptr=stack[top], top=top-1 }
45 PRORR TRVRSL Steps for traversing a tree in preorder sequence are as follows: 1. Visit root 2. Traverse the left subtree 3. Traverse the right subtree
46 Preorder Traversal (ontd.) Perform the preorder traversal of the following tree. Preorder Traversal:
47 LORTM LORTM PRORRTRVRS() { 1. set top=0, stack[top]=null, ptr = root 2. Repeat while ptr!=null 2.1 print ptr -> info 2.2 if (ptr -> right!= NULL) top = top set stack [ top] = ptr -> right 2.3 if ( ptr -> left!= NULL) ptr=ptr -> left else ptr=stack[top], top=top-1 }
48 Postorder Traversal Steps for traversing a tree in postorder sequence are as follows: 1. Traverse the left subtree 2. Traverse the right subtree 3. Visit the root
49 Postorder Traversal (ontd.) Perform the postorder traversal of the following tree. Postorder Traversal:
50 LORTM LORTM POSTORRTRVRS() { 1. set top = 0, stack [top] = NULL, ptr = root 2. Repeat while ptr!=null 2.1 top = top +1, stack [ top ] = ptr 2.2 if (ptr -> right!= NULL) top = top set stack [ top] = - ( ptr -> right ) 2.3 ptr = ptr -> left 3. ptr = stack [top], top = top-1 4. Repeat while ( ptr > 0 ) 4.1 print ptr -> info 4.2 ptr = stack [top], top = top-1 5. if (ptr < 0) 5.1 set ptr = - ptr 5.2 o to step 2 }
51 Recursive Traversal mplementation
52 Just a minute n traversal method, root is processed before traversing the left and right subtrees. nswer: Preorder
53 xpression inary Tree Traversal f an expression is represented as a binary tree, the inorder traversal of the tree gives us an infix expression, whereas the postorder traversal gives us a postfix expression as shown in igure.
54 onstruction of inary Tree iven an order of traversal of a tree, it is possible to construct a tree; for example, consider the folowing order: norder = We can construct the binary trees shown in igure by using this order of traversal
55 onstruction of inary Tree Therefore, we conclude that given only one order of traversal of a tree, it is possible to construct a number of binary trees; a unique binary tree cannot be constructed with only one order of traversal. or construction of a unique binary tree, we require two orders, in which one has to be inorder; the other can be preorder or postorder. or example, consider the following orders: norder = Postorder =
56 onstruction of inary Tree norder = Postorder = We can construct the unique binary tree shown in igure by using these orders of traversal: unique binary tree constructed using its inorder and postorder.
57 Just a minute onstruct the binary tree with the following: norder:- Postorder:-
58 Postorder Traversal (ontd.)
59 Summary n this session, you learned that: tree is a nonlinear data structure that represents a hierarchical relationship among the various data elements. binary tree is a specific type of tree in which each node can have a maximum of two children. inary trees can be implemented by using arrays as well as linked lists, depending upon requirement. Traversal of a tree is the process of visiting all the nodes of the tree once. There are three types of traversals, namely inorder, preorder, and postorder traversal.
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