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

BST Examples

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

Interface Syntax This is an interface

Interface Class that supports an interface

Interface Compiler can verify that class implements an interface Compiler can not verify that implementation is consistent with intent

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

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 = 4 IPL = = 4

Internal Path Length 1 2 N = 4 IPL = = 6 3

Average IPL for N nodes N = 4 Calculate IPL of all possible trees

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

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