Binary Search Trees

1. Binary Search Trees

2. 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

3. 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.

4. Items • Need to map Items to a numerical value • Integers • Value(x) = x • People • Value(x) = ssn • Value(x) = student id

5. 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

6. BST Operations • Constructor • Insert • Find • Findmin • Findmax • Remove

7. 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 }

8. 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 }

9. 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.

10. Removing with value 2 6 6 6 2 8 3 8 3 8 1 1 1 5 5 5 3 3 3 4 4 4

11. 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; }

12. 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

13. 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.

14. 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.

15. Internal Path Length N = 4 IPL = 1 + 1 + 2 = 4 1 1 2

16. Internal Path Length N = 4 IPL = 1 + 2 + 3 = 6 1 2 3

17. 1 1 1 2 2 3 Average IPL for N nodesN = 4 • Calculate IPL of all possible trees 1 2 2

18. 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).

19. 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

20. Do your Lex