lecture 17,18. .pptx

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2. Trees tree : A directed, acyclic structure of linked nodes. directed : Has one-way links between nodes. acyclic : No path wraps back around to the same node twice. binary tree : One where each node has at most two children. Recursive definition: A tree is either: empty ( NULL ), or a root node that contains: data , a left subtree, and a right subtree. (The left and/or right subtree could be empty.) 7 6 3 2 1 5 4 root

3. Trees in computer science folders/files on a computer family genealogy; organizational charts AI: decision trees compilers: parse tree a = (b + c) * d; cell phone T9 d + * a = c b

4. Terminology node : an object containing a data value and left/right children root : topmost node of a tree leaf : a node that has no children branch : any internal node; neither the root nor a leaf parent : a node that refers to this one child : a node that this node refers to sibling : a node with a common subtree : the smaller tree of nodes on the left or right of the current node height : length of the longest path from the root to any node level or depth : length of the path from a root to a given node 7 6 3 2 1 5 4 root height = 3 level 1 level 2 level 3

5. A tree node for integers A basic tree node object stores data and refers to left/right Multiple nodes can be linked together into a larger tree left data right 42 left data right 59 left data right 27 left data right 86

6. Traversals traversal : An examination of the elements of a tree. A pattern used in many tree algorithms and methods Common orderings for traversals: pre-order : process root node, then its left/right subtrees in-order : process left subtree, then root node, then right post-order : process left/right subtrees, then root node 40 81 9 41 17 6 29 m_root

7. Pre order [a+(b-c)]*[(d-e)/(f+g-h)]

8. Pre order traverse As we see the above Diagram according to this picture the pre order traverse is * + a – b c / - d e - + f g h

9. Post order [a+(b-c)]*[(d-e)/(f+g-h)] Traverse a b c - + d e – f g + h –/ *

10. Inorder the ans of INORDERIS D B F E A G C L J H K

11. Algorithm for pre-order PREORD(INFO,LEFT,RIGHT,ROOT) 1-[initialy push the null on STACK, and initialize the PTR] Set top=1,STACK[1]=null and PTR=root 2-repeat the step 3 to 5 while PTR=! NULL 3-apply process to INFO [PTR] 4-if RIGHT[PTR]=!NULL the Set TOP=TOP+1, and STACK[TOP]=RIGHT[PTR] 5-if LEFT[PTR]!=NULL , then Set [PTR= LEFT[PTR] Else Set PTR=STACK[TOP] And TOP=TOP-1 6 EXIT

12. POST ORDER POSTORD(INFO,LEF,RIGHT,ROOT) 1-Push null onto STACK and initialize PTR Set TOP=1,STACK[1]=NULL and PTR=root 2-repeat steps 3 to 5 while PTR!= NULL 3-Set TOP=TOP+1 and STACK[TOP]=PTR 4-if RIGHT[PTR]!=NULL then setTOP=TOP+1 and STACK[TOP]= -RIGHT[PTR] 5-set PTR=LEFT[PTR] 6-setPTR=STACK[TOP] and TOP=TOP-1

13. 7- repaeat while PTR>0 (a)apply process to INFO[PTR] (b)set PTR=STACK[TOP] and TOP=TOP-1 8- if PTR<0 then (a) set PTR= - PTR (b)goto step 2 9 EXIT

14. Post order

15. INORDER Algorithm INORD(INFO,LEFT,RIGHT,ROOT) [push NULL onto STACK and initilize PTR] 1- set TOP=1, STACK[1]=NULL and PTR= ROOT 2-repeat while PTR!=NULL (a)setTOP=TOP+1 and STACK[TOP=PTR] (b)set PTR=LEFT[PTR] 3-ser PTR=STACK[TOP] and TOP=TOP-1 4-repeat step 5 to 7 while PTR!=NuLL 5-apply process to INFO[PTR] 6-IF RIGHT[PTR]!=NULL then (a)set PTR= RIGHT[PTR] (b)goto step 3

16. 7-set PTR=STACK[TOP] and TOP= TOP-1 8-EXIT

17. EXIAMPLE

18. in order TRAVERSING USING STACKS

19. Built a tree: 13,03,55,21,60,33,58,19,96,09,02,62,08

20. Searching & Insertion in binary search trees: Suppose “A” is the given information, For finding “A” in tree or inserting “A” as a new node, the process is given below: Comparing “A” with node “N” 1.if A<N, proceed to left child of N 2.if A>N, proceed to right child of N (b) Repeat step (a) until…… A=N (search is successful) sub tree is empty (search is unsuccessful) (c) insert ”A” at place of empty sub tree.

21. Example:

22. Deletion in binary search trees: Again Suppose “ITEM” is the given information, For finding “ITEM” in tree or inserting “A” as a new node, the process is given below: Comparing “ITEM” with node “N” 1.if ITEM<N, proceed to left child of N 2.if ITEM>N, proceed to right child of N (b) Repeat step (a) until…… ITEM=N (search is successful) sub tree is empty (search is unsuccessful) (c) Delete ”N” ,reset and Exit.

23. Suppose, we want to delete “33” So, ITEM=33 and, initially N=60`

24. Data in form of tables: Before Deletion: 15,25,33,44,50,60,66,75

25. Table after Deletion: 15,25,44,50,60,66,75

26. Table after Deletion: 15,25,44,50,60,66,75