Data Structure and Algorithms Binary Search Tree

introduction,
searching,
insertion and
deletion

Binary Search Trees (BST)
A data structure for efficient searching, insertion
and deletion
Binary search tree property
For every node X
All the keys in its left
subtree are smaller than
the key value in X
All the keys in its right
subtree are larger than the
key value in X

Binary Search Trees
A binary search tree Not a binary search tree

Binary Search Trees
Average depth of a node is O(log N)
Maximum depth of a node is O(N)
The same set of keys may have different BSTs

Searching BST
If we are searching for root (15), then we are done.
If we are searching for a key < root , then we
should search in the left subtree.
If we are searching for a key > root, then we should
search in the right subtree.

Data Structure and Algorithms Binary Search Tree

Searching (Find)
FIND(info, left, right, root, item, loc, par)- finds the item in tree T with root is root and
info, left and right is three array represented in memory. This algorithm returns loc
i.e. location of item and par i.e. parent.
1. [Tree Empty??]
if root==NULL, then set LOC=NULL & PAR=NULL and return.
1. [Item root ??]
If item==INFO[ROOT], then LOC=ROOT & PAR=NULL and return.
1. [Initialize pointer ptr and save]
If item<INFO[ROOT]
then set PTR = LEFT[ROOT] and SAVE=ROOT
Else
set PTR = RIGHT[ROOT] and SAVE=ROOT
[End of if]
1. Repeat 5 and 6 while ptr!=NULL
2. [item found??]
If ITEM=INFO[PTR], then set LOC=PTR and PAR=SAVE, and return.
1. If ITEM<INFO[PTR], then SAVE=PTR and PTR=LEFT[PTR]
Else
Set SAVE=PTR and PTR=RIGHT[PTR]
1. [Search unsuccessful] Set, LOC=NULL and PAR = SAVE
2. Exit
 Time complexity: O(height of the tree)

Sorting: Inorder Traversal of BST
Inorder Traversal of BST prints out all the keys in
sorted order
Inorder: 2, 3, 4, 6, 7, 9, 13, 15, 17, 18, 20

Insertion
Proceed down the tree as you would with a find
If X is found, do nothing (or update something)
Otherwise, insert X at the last spot on the path traversed
Time complexity = O(height of the tree)

Inserting (ADD node)INSBST(info, left, right, root, item, loc, avail)- insert the item in tree
T with root is root and info, left and right is three array
represented in memory. This algorithm returns loc i.e. location of
item or ADD item as new node in tree.
1. Call FIND(INFO, LEFT, RIGHT, ROOT, ITEM, LOC, PAR)
2. If LOC!=NULL, then Exit.
3. [Copy ITEM into new node in AVAIL list]
a) If AVAIL==NULL, Print “OVER FLOW”;
b) Set NEW=AVAIL, AVAIL=LEFT[AVAIL] and
INFO[NEW]=ITEM.
c) Set LOC=NEW,LEFT[NEW]=RIGHT[NEW]=NULL
4. [ADD ITEM to TREE]
If PAR=NULL then, Set ROOT=NEW.
Else IF ITEM<INFO[PAR] , Set LEFT[PAR]=NEW
Else Set RIGHT[PAR]=NEW
1. Exit
Time complexity: O(height of the tree)

Deletion
When we delete a node, we need to consider how we
take care of the children of the deleted node.
This has to be done such that the property of the
search tree is maintained.

Deletion under Different Cases
Case 1: the node is a leaf
Delete it immediately
Case 2: the node has one child
Adjust a pointer from the parent to bypass that node

Deletion Case 3
Case 3: the node has 2 children
Replace the key of that node with the minimum element
at the right subtree
Delete that minimum element
 Has either no child or only right child because if it has a left
child, that left child would be smaller and would have been
chosen. So invoke case 1 or 2.
 Time complexity = O(height of the tree)

Deletion Algorithm
DEL(INFO, LEFT, RIGHT, ROOT, AVAIL, ITEM)
A binary search tree T is in memory, and an ITEM of information is
given. This algorithm delete ITEM from the tree.
1. Call FIND(INFO, LEFT, RIGHT, ROOT, ITEM, LOC, PAR)
2. If LOC=NULL, then write ITEM not in tree and Exit
3. If RIGHT[LOC]!=NULL and LEFT[LOC]!=NULL, then:
Call CASEB(INFO, LEFT, RIGHT, ROOT, LOC, PAR)
Else:
Call CASEA(INFO, LEFT, RIGHT, ROOT, LOC, PAR)
4. Set LEFT[LOC]:=AVAIL and AVAIL :=LOC.
5. Exit

CASEA: only one or, no child
CASEA(INFO, LEFT, RIGHT, ROOT, LOC, PAR)-delete
the Node N at location LOC, where N doesn’t have two
Children. PAR is location of parent node or, PAR=NULL
i.e. ROOT node.
1. [initialize CHILD]
If LEFT[LOC]=NULL and RIGHT[LOC]=NULL, then
CHILD=NULL
Else if LEFT[LOC]!=NULL , then CHILD=LEFT[LOC]
Else CHILD=RIGHT[LOC]
1. If PAR != NULL then: (i.e. NOT A ROOT NODE)
If LOC=LEFT[PAR], then set LEFT[PAR]=CHILD
Else RIGHT[PAR]=CHILD
[End of IF]
Else set ROOT=CHILD.
[End of IF]
1. Exit

CASEB: has 2 children
 CASEB(INFO, LEFT, RIGHT, ROOT, LOC, PAR)-delete the Node N at location
LOC, where N has two Children. PAR is location of parent node or, PAR=NULL i.e.
ROOT node. SUC gives location of inorder successor and PARSUC gives location
of parent of inorder successor .
1. [Find SUC and PARSUC]
a) Set PTR=RIGHT[LOC] and SAVE=LOC
b) Repeat while LEFT[PTR]!=NULL
Set, SAVE=PTR and PTR=LEFT[PTR]
[END OF LOOP]
a) Set SUC=PTR and PARSUC=SAVE.
2. [Delete SUC] Call CASEA(INFO, LEFT, RIGHT, ROOT, SUC,PARSUC)
3. [replace node N by SUC]
a) If PAR != NULL then: (i.e. NOT A ROOT NODE)
If LOC=LEFT[PAR], then set LEFT[PAR]=SUC
Else RIGHT[PAR]=SUC
[End of IF]
Else set ROOT=SUC.
[End of IF]
b) Set, LEFT[SUC]=LEFT[LOC] and
RIGHT[SUC]=RIGHT[LOC]
4. Exit

Data Structure and Algorithms Binary Search Tree

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Data Structure and Algorithms Binary Search Tree