CSC 213 –Large Scale

Programming

Lecture 17:

Binary Search Trees

Today’s Goal

Review the Dictionary ADT Abstracts ideas needed for computer-based

searching (Briefly) examine ways we had implemented

Discuss BinarySearchTree (BST) Relationship to BinaryTree and Heap What BST looks like and how it works Effect on big-Oh notation BST Limitations

Dictionary ADT

dic·tion·ar·y1.Reference book containing collection of words,

with multiple definitions given for each word

2.Book providing many translations into other languages

Dictionary ADT maps key to 1 or more valuesUsed wherever search is (e.g., everywhere)Implementation using Sequence takes O(n) timeWith good batch of hash, could get O(1) time

Binary Search Trees

Store Entrys like Heap

May not be complete Use different ordering

Lower keys in left subtree

Higher keys in right subtree

Equal keys not specified

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SearchRecusive process starting from rootIf root’s key larger,

recurse with left childIf root’s key smaller,

go with right childReturn entry on matchReturn null when not

found (leaf)

Example:TreeSearch(4, root)

Algorithm TreeSearch(Key k, Position p)if isExternal(p)

return pif c.compare(k, p.element().getKey()) < 0

return TreeSearch(k, left(p))else if c.compare(k, p.element().getKey()) == 0

return pelse // k key(v)

return TreeSearch(k, right(p))

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SearchTreeSearch(5, root) Algorithm TreeSearch(Key k, Position p)

if isExternal(p)return p

if c.compare(k, p.element().getKey()) < 0return TreeSearch(k, left(p))

else if c.compare(k, p.element().getKey()) == 0return p

else // k key(v)return TreeSearch(k, right(p))

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Inserting Into Tree

Search for k If k not in tree, expand

leaf where search ended If k already in tree,

Map: Replace node’s entry Dictionary: Keep

searching for next match; only add when external node found

Insert(5)

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Removing an Entry

Starts with search for k Simplest: k not found Simple: node has only

1 external childReplace node with child

Example: remove(4)

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Removing an Entry

Hard: Node has 2 kids Find next node in tree

Go to right child and then keep going left Copy Entry into node Replace descendant

with its right child Example: remove(6)

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BST Performance

Space is linear: O(n) Dictionary methods use

O(h) time h is height of tree

Complete tree is O(log n)

Linked list is O(n) Sadly, this is common

BinaryTree Interface

Tree methods: int size() boolean isEmpty() Iterator<E> iterator() Iterable<E> positions()

Position<E> root() Position<E> parent(p) Iterable<Position<E>> children(p)

boolean isInternal(p) boolean isExternal(p) boolean isRoot(p) E replace (p, e)

Child existence methods: boolean hasLeft(p) boolean hasRight(p)

Child accessor methods: Position<E> left(p) Position<E> right(p)

Rarely call children() on binary trees Much easier to use left()

& right() But must be defined, since

included in interface

Dictionary Interface

Methods searching for Entrys with a given key Entry<K,V> find(K k) Iterator<Entry<K,V>> findAll(K k)

Methods to add or remove data Entry<K,V> insert(K k, V v) Entry<K,V> remove(e)

Iterator methods Iterator<Entry<K,V>> entries() Iterator<V> values()

Collection methods int size() boolean isEmpty()

BST-based Dictionary

BST is a binary tree Should implement BinaryTree interface Could extend existing class or start from scratch Needs methods to insert, remove, & find data

BinaryTree is NOT a Dictionary No similarity between methods Most methods inappropriate for dictionary

So how do we design this?

Using BST for a Dictionary

For best design, implement 2 classes One class for BinaryTree & one for Dictionary Since no methods overlap, very little extra writing Minimum number of methods made public keeping

design simple and clean BST class has add, remove, & find methods

Should work with Entrys Can be used with any Entry-based interface

Dictionary class has field holding BST class Would be easy to modify for ANY search tree

In the Next Lecture

Discuss even better ways of searching

Long awaited AVL Trees Learn what self-balancing means Understand Greg’s love of this search tree

(note: when you do let me know, please)