8.binry search tree

8.binry search tree

• 1.
  Data Structure and Algorithm(CS-102) Ashok K Turuk
• 2.
  Binary Search Tree(BST) Suppose T is a binary tree Then T is called binary search tree if each node N of T has the following property The value at N is greater than every value in the left sub-tree of N and is less than every value in the right sub-tree of N
• 3.
  For any nodey in this subtree key(y) < key(x) x For any node z in this subtree key(z) > key(x)
• 4.
  Binary Search Trees Abinary search tree Not a binary search tree
• 5.
  Binary search trees Averagedepth of a node is O(log N); maximum depth of a node is O(N) Two binary search trees representing the same set:
• 6.
  Searching and Insertingin BST Algorithm to find the location of ITEM in the BST T or insert ITEM as a new node in its appropriate place in the tree [a] Compare ITEM with the root node N of the tree (i) If ITEM < N, proceed to the left child of N (ii) If ITEM > N, proceed to the right child of N
• 7.
  Searching and Insertingin BST Algorithm to find the location of ITEM in the BST T or insert ITEM as a new node in its appropriate place in the tree [b] Repeat Step (a) until one of the following occurs (i) We meet a node N such that ITEM = N. In this case search is successful (ii) We meet an empty sub-tree, which indicates that search is unsuccessful and we insert ITEM in place of empty subtree
• 8.
  Searching BST • Ifwe are searching for 15, then we are done. • If we are searching for a key < 15, then we should search in the left subtree. • If we are searching for a key > 15, then we should search in the right subtree.
• 10.
  Insert 40, 60,50, 33, 55, 11 into an empty BST 40 1. ITEM = 40 40 60 2. ITEM = 60 40 60 50 3. ITEM = 50 40 60 50 33 4. ITEM = 33
• 11.
  Insert 40, 60,50, 33, 55, 11 into an empty BST 40 60 50 33 55 5. ITEM = 55 40 60 50 33 55 11 6. ITEM = 11
• 12.
  Locating an ITEM Abinary search tree T is in memory and an ITEM of information is given. This procedure finds the location LOC of ITEM in T and also the location of the parent PAR of ITEM.
• 13.
  Locating an ITEM Threespecial cases: [1] LOC == NULL and PAR == NULL, tree is empty [2] LOC  and PAR == NULL, ITEM is the root of T [3] LOC == NULL and PAR  NULL, ITEM is not in T and can be added to T as child of the node N with location PAR.
• 14.
  [1] [Tree empty?] If ROOT == NULL, then Set LOC = NULL, PAR = NULL, Exit [2] [ITEM at root ?] If ROOT->INFO == ITEM, then Set LOC = ROOT, PAR = NULL, Exit [3] [Initialize pointer PTR and SAVE] If ITEM < ROOT->INFO then Set PTR = ROOT->LEFT, SAVE = ROOT Else Set PTR = ROOT->RIGHT, SAVE =ROOT FIND(INFO,LEFT,RIGHT,ROOT,ITEM,LOC,PAR)
• 15.
  [4] Repeat Step5 and 6 while PTR  NULL [5] [ITEM Found ?] If ITEM == PTR->INFO, then Set LOC = PTR, PAR = SAVE, Exit [6] If ITEM < PTR->INFO, then Set SAVE = PTR, PTR = PTR->LEFT Else Set SAVE = PTR, PTR = PTR->RIGHT [7] [Search Unsuccessful] Set LOC = NULL, PAR = SAVE [8] Exit
• 16.
  A binary searchTree T is in memory and an ITEM of information is given. Algorithm to find the location LOC of ITEM in T or adds ITEM as a new node in T at location LOC.
• 17.
  [1] Call FIND(INFO, LEFT,RIGHT,ROOT,ITEM,LOC,PAR) [2]If LOC  NULL, then Exit [3] [Copy the ITEM into the node NEW] (a) Create a node NEW (b) NEW->INFO = ITEM ( c) Set LOC = NEW, NEW->LEFT = NULL, NEW->RIGHT = NULL
• 18.
  [4] [Add ITEMto tree] If PAR = NULL Set ROOT = NEW Else If ITEM < PAR->INFO, then Set PAR->LEFT = NEW Else Set PAR->RIGHT = NEW [5] Exit
• 19.
  Binary Search Tree InsertAlgorithm • If value we want to insert < key of current node, we have to go to the left subtree • Otherwise we have to go to the right subtree • If the current node is empty (not existing) create a node with the value we are inserting and place it here.
• 20.
  Binary Search Tree Forexample, inserting ’15’ into the BST?
• 22.
  Deletion from BST Tis a binary tree. Delete an ITEM from the tree T Deleting a node N from tree depends primarily on the number of children of node N
• 23.
  Deletion There are threecases in deletion Case 1. N has no children. N is deleted from the T by replacing the location of N in the parent node of N by null pointer 40 60 50 33 55 11 ITEM = 11
• 24.
  Deletion Case 2. Nhas exactly one children. N is deleted from the T by simply replacing the location of N in the parent node of N by the location of the only child of N 40 60 50 33 55 11 ITEM = 50
• 25.
  Deletion Case 2. 40 60 55 33 11 ITEM =50
• 26.
  Deletion Case 3. Nhas two children. Let S(N) denote the in-order successor of N. Then N is deleted from the T by first deleting S(N) from T and then replacing node N in T by the node S(N) 40 60 50 33 55 11 ITEM = 40
• 27.
  Deletion Case 3. 40 60 55 33 11 Delete 50 50 60 55 33 11 Replace40 by 50
• 28.
  Types of BinaryTrees • Degenerate – only one child • Balanced – mostly two children • Complete – always two children Degenerate binary tree Balanced binary tree Complete binary tree
• 29.
  Binary Trees Properties •Degenerate – Height = O(n) for n nodes – Similar to linear list • Balanced – Height = O( log(n) ) for n nodes – Useful for searches Degenerate binary tree Balanced binary tree
• 30.
  Binary Search Trees •Key property – Value at node • Smaller values in left subtree • Larger values in right subtree – Example • X > Y • X < Z Y X Z
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  Binary Search Trees •Examples Binary search trees Non-binary search tree 5 10 30 2 25 45 5 10 45 2 25 30 5 10 30 2 25 45
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  Example Binary Searches •Find ( 2 ) 5 10 30 2 25 45 5 10 30 2 25 45 10 > 2, left 5 > 2, left 2 = 2, found 5 > 2, left 2 = 2, found
• 33.
  Example Binary Searches •Find ( 25 ) 5 10 30 2 25 45 5 10 30 2 25 45 10 < 25, right 30 > 25, left 25 = 25, found 5 < 25, right 45 > 25, left 30 > 25, left 10 < 25, right 25 = 25, found
• 34.
  Binary Search Properties •Time of search – Proportional to height of tree – Balanced binary tree • O( log(n) ) time – Degenerate tree • O( n ) time • Like searching linked list / unsorted array • Requires – Ability to compare key values
• 35.
  Binary Search TreeConstruction • How to build & maintain binary trees? – Insertion – Deletion • Maintain key property (invariant) – Smaller values in left subtree – Larger values in right subtree
• 36.
  Binary Search Tree– Insertion • Algorithm 1. Perform search for value X 2. Search will end at node Y (if X not in tree) 3. If X < Y, insert new leaf X as new left subtree for Y 4. If X > Y, insert new leaf X as new right subtree for Y • Observations – O( log(n) ) operation for balanced tree – Insertions may unbalance tree
• 37.
  Example Insertion • Insert( 20 ) 5 10 30 2 25 45 10 < 20, right 30 > 20, left 25 > 20, left Insert 20 on left 20
• 38.
  Binary Search Tree– Deletion • Algorithm 1. Perform search for value X 2. If X is a leaf, delete X 3. Else // must delete internal node a) Replace with largest value Y on left subtree OR smallest value Z on right subtree b) Delete replacement value (Y or Z) from subtree Observation – O( log(n) ) operation for balanced tree – Deletions may unbalance tree
• 39.
  Example Deletion (Leaf) •Delete ( 25 ) 5 10 30 2 25 45 10 < 25, right 30 > 25, left 25 = 25, delete 5 10 30 2 45