Threaded binary tree
- 2. Content:- • Introduction • Representation of Binary tree • Advantages of binary tree • Various operations • InorderSuccessor • InorderPredecessor • InorderTraversal • Insertion • Deletion 2
- 3. Let’s start with the first set of slides 3 Introduction:- The fact that binary tree more than 50 percent of link field are with null values, thereby wasting the memory space. A clever way to utilize these Null-link field has been devised by perlis and Thornton.There idea was to store pointers of some nodes in these link field; These extra pointers are called threads and the tree is specially termed threaded binary tree
- 4. 4 Representation of threaded binary tree:- We have to first decide to which node threaded should point. In fact there are three ways to threaded binary tree, these correspondto in order, preorder, postorder traversal The threaded binary tree corresponding to inorder traversal is called Inorder threating and that corresponding to preorder traversal is called preorder threading similar to postorder threading
- 5. If any node N has its right(left) link Field empty, then this field will be initalized by the inorder successor 5
- 6. 6
- 7. The data content of this header node is null and this node can be considered as the inorder predecessor and the inorder successor of the first and the last node respectively in the picture shows an empty inorder threaded binary tree. The complete form of an inorder threaded binary tree is shown in the picturein case of an empty inorder threaded binary tree, the left link field field of the headér node is a threaded which point it’s self 7
- 8. A simple node structure have LCHILD and RCHILD is not sufficient. The noade structure that is required is showing inthe picture, this node structure includes two extra field:LTAG and RTAG. They are binary fields and will stroe their TRUE Or FALSE. LTAG=TRUE the pointer of LCHILD is a threaded LTAG=FALSE the pointer of LCHILD is a link RTAG=TRUE the pointer of RCHILD is a threaded RTAG=FALSE the pointer of RCHILD is a link 8
- 9. 9 tree The traversal operations is faster than that of its unthreaded version, because with a threaded binary tree, non recursive implementation is possible that can run faster and doesn’t require of stack management. We can efficiently determine the predecessor and successor nodes starting from any node. In case of on an unthreaded binary tree, however, this task is more Time consuming And difficult.
- 10. 10 Any node can be accessed From any other node. Threads are usually more to upward whereas link are downward. Although insertion into and deletion from a threaded tree are time consuming operation these are very esay to implement.
- 11. 11 Various operations on threaded binary tree:- • To find inorder successor of any node • To find inorder predecessor of any node • To find inorder traversal of any node • Insertion of a node into a threaded binary tree • Deletion of a node from a threaded binary tree
- 12. To find the inorder successor of any node:- the address of any node in an inorder successor is very straightforward. If N does not have a right child then the thread Points to it’s inorder successor Else If N has a right sub tree then the left most node in this right sub tree which does not have any left child is the
- 13. 13 The given picture inorder is BAGEDFC Inorder successor of ‘B’is ‘A’ Inorder successor of ‘A’is ‘G’ Inorder successor of ‘D’ is ‘F’ Inorder successor of ‘C’ is the HEATER We can formulate the procedure Inordersuccessor(N) to find the inorder successor of any node N in a threaded binary tree
- 14. 14 Algorithm Inordersuccessor:- Input:PTR is the pointer to any node whose inorder successor has to be found Output:-pointer to the inorder successor of the node PTR Data structure:-Linked structure of Threaded binary tree Steps:- succ=PTR->RCHILD If(PTR->RTAG=FALSE)then While(succ->LTAG=FALSE) Succ=succ->LCHILD EndWhile EndIF Return(succ) stop
- 15. 15 Algorithm Inorderpredecessor:- Input:PTR is the pointer to any node whose inorder Predecessor has to be found Output:-pointer to the inorder Predecessor of the node PTR Data structure:-Linked structure of Threaded binary tree Steps:- Pred=PTR->LCHILD If(PTR->LTAG=FALSE)then While(Pred->RTAG=FALSE) Pred=Pred->RCHILD EndWhile EndIF Return(pred) stop
- 16. 16 Algorithm InorderTraversal_TBT:- Input:An inorder threaded binary tree Output:-Inorder traversal of the threaded binary tree Data structure:-Linked structure of Threaded binary tree having HEADER as the pointer to root Steps:- ptr=HEADER While(TRUE) do ptr=Inordersuccessor (ptr) If(ptr=HEADER) Exit Else Process (ptr) EndIF EndWhile stop
- 17. Insertion of a node into a threaded binary tree 17
- 18. 18 Add a new node at Right child node NO Right sub tree
- 19. 19 Add a new node at Left child node NO Left sub tree
- 20. 20 Add a new node at Right child node it has non empty Right sub tree
- 21. 21 Add a new node at Leftt child node it has Left sub tree
- 22. 22
- 23. 23
- 24. 24
- 25. 25
- 26. 26 Algorithm InsertInorder_TBT:- Inputs:- (a) HEADER, the pointer to the header node of the threaded binary tree (b) X is the data of a node after which the insertion has to be done (c) N is the data node to be interested Output:-If X exists in the tree then N is Inserted after X. Data structure:- Linked structure of threaded binary tree. Steps:- ptr=HEADER->LCHILD flag=FALSE While(ptr! =HEADER) and(flag=FALSE)
- 27. 27 Xptr=ptr Flag=TRUE Else ptr=Inordersuccessor (ptr) Endif EndWhile If(flag=FALSE) then Print”Node doesn’t exist:no insertion “ Exit EndIf Read option=L/R for Left(L) or Right (R) child Nptr=GetNode (NODE) Case: option=‘L’ If(Xptr->LTAG=TRUE) then Nptr->LCHILD=Xptr->LCHILD Nptr->LTAG=TRUE Xptr->LCHILD=Nptr Xptr->LTAG=FALSE Nptr-RCHILD=Xptr Nptr->RTAG=TRUE
- 28. Else Lptr=Xptr->LCHILD Xptr->LCHILD=Nptr Xptr->LTAG=FALSE Nptr->RCHILD=Xptr Nptr->RTAG=TRUE Nptr->LCHILD=Lptr Nptr->LTAG=FALSE ptr=InorderPredecessor (Xptr) ptr->RCHILD=Nptr End If Case: option=‘R’ If(Xptr->RTAG=TRUE) then Nptr->RCHILD=Xptr->RCHILD Nptr->RTAG=TRUE Xptr->RCHILD=Nptr Xptr->RTAG=FALSE Nptr->LCHILD=Xptr Nptr->LTAG=TRUE 28
- 30. 30 Deletion of a node from a threaded binary tree
- 31. 31 Delete a leaf node right or left
- 32. 32 Delete of node having right sub tree or left sub tree
- 33. 33 Delete of the node having both sub tree
- 34. 34
- 35. 35
- 36. Algorithm Parent:- Input:- The pointer PTR of a node whose parent is to be found out. Output:-The pointer PARENT node of a given node PTR Data Structure:- Linked structure of a threaded binary tree. Steps:- varptr=PTR While(varptr, RTAG=0) do varptr=varptr->RCHILD End while pred=varptr->RCHILD If(pred->LCHILD=ptr)then parent=pred Else pred=pred ->LCHILD While (pred->LCHILD! =ptr) Pred=pred->RCHILD End while parent=pred End if Return(parent) Stop 36
- 37. deleted Output:- The threaded binary search tree without a node PTR or it’s data content Data structure:-Linked structure of threaded binary tree Steps: parent=Parent (PTR) If(Ptr->LTAG=TRUE) and(PTR->RTAG=TRUE) then Case=1 Else If(PTR->LTAG=FALSE) and(PTR->RTAG=FALSE) then Case=3 Else Case=2 End if End if 37
- 38. Do case=1 If(parent->RCHILD=PTR) then parent->RCHILD=PTR->RCHILD parent->RTAG=TRUE Else If(parent->LCHILD=PTR) then parent->LCHILD=PTR->LCHILD parent->LTAG=TRUE End if End if ReturnNode(PTR) Exit Endo Do case=2 // Find the sub tree of the node PTR If(PTR->RTAG=FALSE) then child=PTR->RCHILD Else If(PTR->LTAG=FALSE) then Child=PTR->LCHILD End if 38
- 39. //modify the link field of parent If(parent->RCHILD=PTR) then parent->RCHILD=child Else If(parent->LCHILD=PTR) then parent->LCHILD=child Endif //modify the thread pointer in succeding node of PTR If(PTR->RTAG=FALSE) then Succ=inorder successor (PTR) Succ->LCHILD=parent Else If(PTR->LTAG=FALSE) then Pred=inorderPredecessor (PTR) Pred->RCHILD=parent Endif Endif Return node (PTR) Exit End Do 39
- 40. Do case=3 Succ=Inorder successor (PTR) PTR->DATA=succ->DATA Delete inorder_TBST(succ) End Do Stop 40
- 41. Thankyou 41