Chapter 9 contd.Binary Search Trees

Anshuman RazdanDiv of Computing Studies

razdan@asu.eduhttp://dcst2.east.asu.edu/~razdan/cst230/

CST 230 Razdan et al 2

BST and Total Order• Binary search trees are a subset (generalization) of

Binary Trees• The set of objects stored in a binary tree must

come from a totally ordered set• A Total Order has the following mathematical

properties:– Equality– Totality– Consistency– Transitivity

CST 230 Razdan et al 3

BST Definition

• In a Binary Search Tree, the elements of the nodes can be compared with a total order semantics. These two rules are followed for every node n:– Every element in n’s left subtree is less than or

equal to the element in n– Every element in n’s right subtree is greater

than or equal to the element in n.

CST 230 Razdan et al 4

Example

45

9 53

3 17

20

53 54

CST 230 Razdan et al 5

Typical BST Methods

• add

• remove

• find

• other variations:– find min– find max– add/combine BSTs

CST 230 Razdan et al 6

add a new Node

• Suppose we want to add 23 and 50 to the following BST

45

9 53

3 17

20

53 54

CST 230 Razdan et al 7

add cont...if tree is empty

make the new node the rootelse insert in the subtree under the current root

if val <= rootif root has left child

insert in subtree under left childelse

make val the left childelse

if root has right childinsert in subtree under right child

elsemake val the right child

CST 230 Razdan et al 8

remove

• Suppose we want to remove 17, 45 from the BST

45

9 53

3 17

20

53 54

CST 230 Razdan et al 9

remove cont...

find node to removeif node is a leaf

set parent’s appropriate child to nullelse if node has no left child

set parent’s appropriate child to right childelse if node has no right child

set parent’s appropriate child to left childelse

set data in node to max value in left subtreeremove max value in left subtree

CST 230 Razdan et al 10

Complexity of BST methods

• Best Case– find

– add

– remove

• Worst Case– find

– add

– remove