![Binary Tree ADT
A binary tree T is defined as a finite set of elements, called nodes,
such that:
T is empty [called the null tree or empty tree]
T contains a distinguished node R, called the root of T form
an ordered pair of disjoint binary trees LT and RT.
If T does contain a root R.
Two trees LT and RT are called respectively, the left and right sub
trees of R.
If LT is non-empty, then its root is called the left successor of R.
If RT is non-empty, then its root is called the right successor of R.
Binary Tree](https://image.slidesharecdn.com/trees-250118171930-094dd09b/85/Trees-Basic-Terminology-and-Binary-Trees-12-320.jpg)
Trees, Basic Terminology and Binary Trees
- 1. DATA STRUCTURES AND ALGORITHMS Mrs.V.JAYAVANI., M.S(IT&M) Assistant Professor Department of Computer Science UNIT – III TREES
- 2. Tree ADT Tree Traversals Binary Tree ADT Expression Trees Applications of Trees Binary Search Tree ADT SUMMARY Threaded Binary Trees AVL Trees B-Tree B+ Tree Heap Applications of heap
- 3. So far we have discussed mainly linear data structures – Arrays, Lists, Stacks and Queues. Now we will discuss a non-linear data structure called tree.
- 4. TREE A tree is a non-linear and hierarchical data structure that has a group of nodes. Tree is a sequence of nodes. Trees represent data containing a hierarchical relationship between elements. Eg: Records, Family Trees and Table Contents. Consider a parent-child relationship.
- 6. Basic Terminology Node – Node is elements of a Tree. (A,B,C,D,E, F,G,H,I,J) Root Node − The first node is called as Root node. Every tree must have only one root node. (A is a root node.) Edge – Edge is a connection between one node to another. In a tree with ‘N’ number of nodes there will be a maximum of ‘N-1’ number of edges. No. of Edges = N – 1 = 10 – 1 = 9 Edges Parent Node − The node which is a predecessor of any node is called as Parent Node. It can also be defined as “the node which has child / children”. Here, A, B, C, D and E are Parent nodes.
- 7. Basic Terminology Child − The node which is descendant of any node is called as child node. In a tree all the nodes except root are child nodes. Here, B,C,D,E,F,G,H,I and J are children nodes. Siblings – The nodes with the same parent are called sibling nodes. Here, B & C, D & E, F & G and H & I are Siblings Leaf − The node which does not have any child is called as leaf node. It is also called as External nodes (or) Terminal nodes. Here, H, I, J, F and G are Leaf nodes. Internal Node – The node which has atleast one child is called as Internal node. Nodes other than leaf nodes are called as internal nodes. Here, A, B, C, D and E are Internal nodes Degree – The degree of a node is total number of children it has. The highest degree of a node among all the nodes in a tree is called as “Degree of Tree”. Here, Degree of A is 2, Degree of D is 2, Degree of F is 0.
- 8. Basic Terminology Levels − In a tree, each step from top to bottom is called as a level. Here, Level 0 is A, Level 1 is B & C, Level 2 is D,E,F, & G Path − It is a sequence of nodes and edges between two nodes. Here, ‘Path’ between A & I is A – B – D - I Height − Height of node is total number of edges from leaf to that node is longest path. Here, Height of Tree is 3. Depth − Depth of node is total number of edges from root to that node. Here, Depth of tree is 3. Sub Tree – Each child from a node forms a sub tree recursively. Every child node will form a sub tree on its parent node. Here, D,H & I is sub tree.
- 9. Tree Traversals Traversal is a process to visit all the nodes of a tree and may print their values too. All nodes are connected via edges (links) we always start from the root (head) node. There are three ways which we use to traverse a tree Pre-order Traversal In-order Traversal Post-order Traversal
- 10. Pre-order, in-order, post-order In-order <left><root><right> (HDIBJEAFCG) I. Traverse the left sub tree of R in preorder. II. Process the root R. III. Traverse the right sub tree of R in preorder. Pre-order <root><left><right> (ABDHIEJCFG) I. Process the root R. II. Traverse the left sub tree of R in preorder. III. Traverse the right sub tree of R in preorder. Post-order <left><right><root> (HIDJEBFGCA) I. Traverse the left sub tree of R in preorder. II. Traverse the right sub tree of R in preorder. III. Process the root R.
- 11. Example Consider the traversal of a tree Preorder : ABCEIFJDGHKL Inorder: EICFJBGDKHLA which of the following is correct post order traversal? Solution: Post Order traversal is IEJFCGKLHDBA 2 1 5 3 4 6 7
- 12. Binary Tree ADT A binary tree T is defined as a finite set of elements, called nodes, such that: T is empty [called the null tree or empty tree] T contains a distinguished node R, called the root of T form an ordered pair of disjoint binary trees LT and RT. If T does contain a root R. Two trees LT and RT are called respectively, the left and right sub trees of R. If LT is non-empty, then its root is called the left successor of R. If RT is non-empty, then its root is called the right successor of R. Binary Tree
- 13. Binary Tree ADT T consists of 10 nodes, represented by the letter A through J. The root of T is the node A at the top of the diagram. B is a left successor and C is a right successor of that node A. The left subtree of he root A consists of the nodes B, D, E, H & I. The right subtree of he root A consists of the nodes C, F, G & J. Any node N in a binary tree T has either 0,1 or 2 successor. The nodes A, B, C and E have two successors. The node G have only one successor. The nodes D, H, I, F and J have no successors. The nodes with no successors are called Terminal nodes. Binary Tree
- 14. Expression Trees An expression tree is a tree built up from the simple operands as the leaves of binary tree and operators as the non -leaves of binary tree. It is a special kind of binary tree in which:- (i) Each leaf node contains single operand. (ii) Each non-leaf node contains a single binary operator. (iii) The left and right subtrees of an operator node represent sub-expression that must be evaluated before applying the operator at the root of the subtree. The levels in a binary expression tree represent the precedence of operators. The operators at the lower level must be evaluated first and then the operators at the next level and so on and at the last operator at the root node is applied and there by the expression is evaluated.
- 15. Expression Trees Eg. Expression: - (a-b) * (c+ d ) Prefix expression is *-ab+cd similar to preorder traversal Postfix expression is ab-cd+* similar to postorder traversal * + - a b c d
- 16. Construction of an Expression Tree (1) From prefix and infix expression 1. Read the prefix expression, the first element becomes the root. 2. Now scan the infix expression till you get an element found in step1. Place all the elements left of this element to the left of it and other element to right of it. 3. Repeat steps 1 and 2 till all the elements from infix expression gets placed in tree. 4. stop
- 17. Construction of an Expression Tree (2) From infix and postfix expression 1. Read the postfix expression, the last element becomes the root. 2. Now scan the infix expression till you get an element found in step1. Place all the elements left of this element to the left of it and other element to right of it. 3. Repeat steps 1 and 2 till all the elements from infix expression get placed in tree. 4. Stop
- 18. Construction of an Expression Tree (3) From a given postfix expression 1. Read the element from postfix expression from left to right. 2. If it is an operand then push address of this node onto stack go to step 1. 3. If the element is an operator then pop twice which gives the address of the two operands. Create a new node for this operator and attach the operand to the left and right branch. Push the address of this node onto stack . 4. If all the elements from postfix expression are not read then go to step 1. 5. Pop the contents of stack which give the root address of expression tree. 6. Stop
- 19. Example Construct expression tree from following expressions:- Postfix expression:- 5,3,2,^,*,3,7,3,+,10,/,+/ Infix expression :- (5*3^2)/(3+(7+3)/10) Sol. From postfix we get root i.e. / ,hence (5*3^2) is left subtree and (3+(7+3)/10) is right subtree. Now again find root from postfix for right You get + , now 3 is left and is (7+3)/10 is right, then 3 is left child. In right subtree / is root , (7+3) is left subtree and 10 is right subtree. 10 is root for right subtree. Now in left subtree + is root and 7is left child and 3 is right child. Similarly for left subtree * is root and 5 is left subtree and 3^2 is right subtree. Now again from right subtree root is ^, 3 is left child and 2 is right child. 3 2 3 7 + 10 / 3 Λ 5 + * /
- 20. Applications of Trees Directory structure of a file store Structure of an arithmetic expressions Used in almost every 3D video game to determine what objects need to be rendered. Used in almost every high-bandwidth router for storing router-tables. Used in compression algorithms, such as those used by the .jpeg and .mp3 file- formats.