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A generic binary search tree II let rec find_max = function Empty -> assert false Node (_, x, Empty) -> x Node (_, x, r) -> find_max r;; # val find_max : 'a bst 

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[PDF] Topic 19 Binary Search Trees

Expands on the binary search technique and A binary search tree is a binary tree in which every node's left use structural recursion and polymorphism

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Topic 19

Binary Search Trees

"Yes. Shrubberies are my trade. I am a shrubber. My name is 'Roger the Shrubber'. I arrange, design, and sell shrubberies." -Monty Python and The Holy Grail

CS3142

The Problem with Linked Lists

Accessing a item from a linked list takes

O(N) time for an arbitrary element

Binary trees can improve upon this and

reduce access to O(log N) time for the average case

Expands on the binary search technique and

allows insertions and deletions

Worst case degenerates to O(N) but this can

be avoided by using balanced trees (AVL,

Red-Black)

Binary Search Trees

CS3143

Binary Search Trees

A binary search tree is a binary tree in which every node's left subtree holds values less than the node's value, and every right subtree holds values greater than the node's value.

A new node is added as a leaf.

parent left childright child root 17 1119
< 17> 17

Binary Search Trees

BST Insertion

Add the following values one at a time to an

initially empty binary search tree using the simple algorithm:

90 20 9 98 10 28 -25

What is the resulting tree?

CS314Binary Search Trees4

Traversals

What is the result of an inorder traversal of

the resulting tree?

How could a preorder traversal be useful?

CS314Binary Search Trees5

Clicker 1

After adding N distinct elements in random

order to a Binary Search Tree what is the expected height of the tree? (using the simple insertion algorithm)

A.O(logN)

B.O(N1/2)

C.O(N)

D.O(NlogN)

E.O(N2)

CS3146Binary Search Trees

Clicker 2

After adding N distinct elements to a Binary

Search Tree what is the worst case height

of the tree? (using the simple insertion algorithm)

A.O(logN)

B.O(N1/2)

C.O(N)

D.O(NlogN)

E.O(N2)`

CS3147Binary Search Trees

CS3148

Worst Case Performance

Insert the following values into an initially

empty binary search tree using the simple, naïve algorithm:

2 3 5 7 11 13 17

What is the height of the tree?

What is the worst case height of a BST?

Binary Search Trees

CS3149

Node for Binary Search Trees

public class BSTNode { private Comparable myData; private BSTNode myLeft; private BSTNode myRightC; public BinaryNode(E item) {myData = item;} public E getValue() {return myData;} public BinaryNode getLeft() {return myLeft;} public BinaryNode getRight() {return myRight;} public void setLeft(BSTNode b) {myLeft = b;} // setRight not shown }Binary Search Trees

CS31410

More on Implementation

Many ways to implement BSTs

Using nodes is just one and even then many

options and choices public class BinarySearchTree> { private BSTNode root; private int size;

Binary Search Trees

CS31411

Add an Element, Recursive

Binary Search Trees

CS31412

Add an Element, Iterative

Binary Search Trees

Clicker 3

What are the best case and worst case order

to add N distinct elements, one at a time, to an initially empty binary search tree using the simple add algorithm?

BestWorst

A.O(N)O(N)

B.O(NlogN)O(NlogN)

C.O(N)O(NlogN)

D.O(NlogN)O(N2)

E.O(N)O(N2)

13 // given int[] data // no duplicates in // data

BST b =

new BST(); for(int x : data) b.add(x);

CS31414

Performance of Binary Trees

For the three core operations (add, access,

remove) a binary search tree (BST) has an average case performance of O(log N)

Even when using the naïve insertion /

removal algorithms no checks to maintain balance balance achieved based on the randomness of the data inserted

Binary Search Trees

CS31415

Remove an Element

Five (four?) cases

not present node is a leaf, 0 children (easy) node has 1 child, left or right (easy) node has 2 children ("interesting")

Binary Search Trees

CS31416

Properties of a BST

The minimum value is in the left

most node

The maximum value is in the right

most node useful when removing an element from the BST

Binary Search Trees

CS31417

Alternate Implementation

In class examples of dynamic data structures

have relied on null terminated ends.

Use null to show end of list or no children

Alternative form

use structural recursion and polymorphism

Binary Search Trees

CS31418

BST Interface

publicinterfaceBSTComparable>{ publicintsize(); publicbooleancontains(E obj); publicBST add(E obj);

Binary Search Trees

CS31419

EmptyBST

publicclassEmptyBST> implementsBST { privatestaticfinal EmptyBST theOne= newEmptyBST(); privateEmptyBST() {} publicstaticEmptyBST getEmptyBST(){ returntheOne; } publicBST add(E obj) { return new NEBST(obj); } publicbooleancontains(E obj) { returnfalse; } publicintsize() { return0; } }Binary Search Trees

CS31420

Non Empty BST Part 1

publicclassNEBST >implementsBST { privateE data; privateBST left; privateBST right; publicNEBST(E d) { data = d; right = EmptyBST.getEmptyBST(); left = EmptyBST.getEmptyBST(); publicBST add(E obj) { intdirection = obj.compareTo( data ); if ( direction < 0 ) left = left.add( obj ); elseif ( direction > 0 ) right = right.add ( obj ); returnthis;

Binary Search Trees

CS31421

Non Empty BST Part 2

publicbooleancontains(E obj){ intdir = obj.compareTo(data); if( dir == 0 ) returntrue; elseif(dir < 0) returnleft.contains(obj); else returnright.contains(obj); publicintsize() { return1 + left.size() + right.size(); }Binary Search Treesquotesdbs_dbs19.pdfusesText_25