Binary Search Trees

BST Properties

• Have all properties of binary tree

• Items in left subtree are smaller than items in any node

• Items in right subtree are larger than items in any node

Items

• Items must be comparable

• All items have a unique value

• Given two distinct items x and y either– value(x) < value(y)– value(x) > value(y)

• If value(x) = value(y) then x = y

• It will simplify programming to assume there are no duplicates in our set of items.

Items

• Need to map Items to a numerical value

• Integers– Value(x) = x

• People– Value(x) = ssn– Value(x) = student id

Comparable Interface

• Want general tree code• Requirement of item is that it supports

– <– >– =

• Java uses Interfaces for implementation• Similar to abstract method• Specify a method that using class is

responsible for

BST Operations

• Constructor

• Insert

• Find– Findmin– Findmax

• Remove

BST Operations

• Generally Recursive

BinaryNode operation( Comparable x, BinaryNode t ) {// End of path

if( t == null ) return null;

if( x.compareTo( t.element ) < 0 )return operation( x, t.left );

else if( x.compareTo( t.element ) > 0 )return operation( x, t.right );

else return t; // Match}

BST Find Method

private BinaryNode find( Comparable x, BinaryNode t ) { if( t == null ) return null; if( x.compareTo( t.element ) < 0 ) return find( x, t.left ); else if( x.compareTo( t.element ) > 0 ) return find( x, t.right ); else return t; // Match }

BST Remove Operations

• Remove– Node is leaf

• Remove node

– Node has one child• Replace node with child

– Node has two children• Replace node with smallest child of right subtree.

Removing with value 2

6

2 8

1 5

3

4

6

3 8

1 5

3

4

6

3 8

1 5

3

4

Remove method private BinaryNode remove( Comparable x, BinaryNode t ) { if( t == null ) return t; // Item not found; do nothing if( x.compareTo( t.element ) < 0 ) t.left = remove( x, t.left ); else if( x.compareTo( t.element ) > 0 ) t.right = remove( x, t.right ); else if( t.left != null && t.right != null ) // Two children { t.element = findMin( t.right ).element; t.right = remove( t.element, t.right ); } else t = ( t.left != null ) ? t.left : t.right; return t; }

Internal Path Length

• Review depth/height

• Depth– Depth is number of path segments from root

to node– Depth of node is distance from root to that

node.– Depth is unique– Depth of root is 0

Internal Path Length

• Height– Height is maximum distance from node to a

leaf.– There can be many paths from a node to a

leaf.– The height of the tree is another way of

saying height of the root.

Internal Path Length

• IPL is the sum of the depths of all the nodes in a tree

• It gives a measure of how well balanced the tree is.

Internal Path Length

11

2

N = 4IPL = 1 + 1 + 2 = 4

Internal Path Length

1

2

N = 4IPL = 1 + 2 + 3 = 6

3

Average IPL for N nodesN = 4

• Calculate IPL of all possible trees

1

2

3

11

2

1

2 2

BST Efficiency

• If tree is balanced O(log(n))

• No guarantee that tree will be balanced

• Analysis in book suggests on IPL = O(nlog(n))

• This analysis is based on the assumption that all trees are equally likely

• Could always get the worst case (a degenerate tree).

Where do BST fit in

• Simple to understand

• Works for small datasets

• Basis for more complicated trees

• Using inheritance can implement– AVL trees– Splay trees– Red Black trees

Do your Lex