CSE 326: Data StructuresBinary Search Trees

Ben Lerner

Summer 2007

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Tree CalculationsRecall: height is max

number of edges from root to a leaf

Find the height of the tree...

t

Can you do this in less than O(n) time?

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Tree Calculations Example

A

E

B

D F

C

G

IH

KJ L

M

L

N

How high is this tree?

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Binary Trees• Binary tree is

– a root– left subtree (maybe

empty) – right subtree (maybe

empty)

• Representation:

A

B

D E

C

F

HG

JI

Data

right pointer

leftpointer

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Binary Tree: RepresentationA

right pointer

leftpointer A

B

D E

C

F

Bright

pointerleft

pointer

Cright

pointerleft

pointer

Dright

pointerleft

pointer

Eright

pointerleft

pointer

Fright

pointerleft

pointer

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Binary Tree: Special Cases

A

B

D E

C

GF

IH

A

B

D E

C

F

A

B

D E

C

GF

Full Tree

Complete Tree Perfect Tree

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Binary Tree: Some Numbers!For binary tree of height h:

– max # of leaves:

– max # of nodes:

– min # of leaves:

– min # of nodes:

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More Recursive Tree Calculations:

Tree TraversalsA traversal is an order for

visiting all the nodes of a tree

Three types:• Pre-order: Root, left subtree, right

subtree

• In-order: Left subtree, root, right subtree

• Post-order: Left subtree, right subtree, root

+

*

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(an expression tree)

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Traversals

void traverse(BNode t){

if (t != NULL)

traverse (t.left);

print t.element;

traverse (t.right);

}

} Which kind of traversal is this?

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ADTs Seen So Far

• Stack– Push– Pop

• Queue– Enqueue– Dequeue

What about decreaseKey?

• Priority Queue– Insert– DeleteMin

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The Dictionary ADT

• Data:– a set of

(key, value) pairs

• Operations:– Insert (key,

value)– Find (key)– Remove (key) The Dictionary ADT is sometimes

called the “Map ADT”

• blernerBen Lerner,OH: Tues, 10:45—12:00CSE 212

• ioannisIoannis Giotis,OH: M 11:00CSE 430

• gyanitGyanit Singh,OH: M 11:00CSE 430

insert(blerner, ….)

find(ioannis)• ioannis Ioannis Giotis, …

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A Modest Few Uses

• Sets

• Dictionaries

• Networks : Router tables

• Operating systems : Page tables

• Compilers : Symbol tables

Probably the most widely used ADT!

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Implementations

• Unsorted Linked-list

• Unsorted array

• Sorted array

insert deletefind

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Binary Search Tree Data Structure

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• Structural property– each node has 2 children– result:

• storage is small• operations are simple• average depth is small

• Order property– all keys in left subtree smaller

than root’s key– all keys in right subtree larger

than root’s key– result: easy to find any given key

• What must I know about what I store?

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Example and Counter-Example

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21BINARY SEARCH TREE NOT ABINARY SEARCH TREE

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Find in BST, RecursiveNode Find(Object key, Node root) { if (root == NULL) return NULL;

if (key < root.key) return Find(key, root.left); else if (key > root.key) return Find(key, root.right); else return root;}

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Runtime:

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Find in BST, IterativeNode Find(Object key, Node root) {

while (root != NULL && root.key != key)

{ if (key < root.key) root = root.left; else root = root.right; }

return root;}

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Runtime:

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Insert in BST

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Runtime:

Insert(13)Insert(8)Insert(31)

Insertions happen only at the leaves – easy!

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BuildTree for BST• Suppose keys 1, 2, 3, 4, 5, 6, 7, 8, 9 are inserted

into an initially empty BST. Runtime depends on the order!

– in given order

– in reverse order

– median first, then left median, right median, etc.

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Bonus: FindMin/FindMax

• Find minimum

• Find maximum2092

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Deletion in BST

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Why might deletion be harder than insertion?

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Delete Operation

• Delete is a bit trickier…Why?• Suppose you want to delete 10• Strategy:

– Find 10– Delete the node containing 10

• Problem: When you delete a node,what do you replace it by?

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Delete Operation

• Problem: When you delete a node,what do you replace it by?

• Solution:– If it has no children, by NULL– If it has 1 child, by that child– If it has 2 children, by the node with

the smallest value in its right subtree(the successor of the node)

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Delete “5” - No children

Find 5 node

Then Freethe 5 node and NULL the pointer to it

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Delete “24” - One child

Find 24 node

Then Freethe 24 node and replace the pointer to it witha pointer to itschild

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Delete “10” - two children

Find 10,Copy the smallestvalue inright subtreeinto the node

Then recursivelyDelete node with smallest valuein right subtreeNote: it does nothave two children

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Delete “11” - One child

Remember11 node

Then Freethe 11 node and replace the pointer to it witha pointer to itschild

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Runtimes

• Find? Insert? Delete?

• What is the average height of a BST?

• What is the maximum height?

• What happens when we insert nodes in sorted order?

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Balanced BST

Observation• BST: the shallower the better!• For a BST with n nodes

– Average height is O(log n)– Worst case height is O(n)

• Simple cases such as insert(1, 2, 3, ..., n)lead to the worst case scenario

Solution: Require a Balance Condition that1. ensures depth is O(log n) – strong enough!2. is easy to maintain – not too strong!

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Potential Balance Conditions1. Left and right subtrees of the root

have equal number of nodes

2. Left and right subtrees of the roothave equal height

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Potential Balance Conditions3. Left and right subtrees of every node

have equal number of nodes

4. Left and right subtrees of every nodehave equal height

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The AVL Balance ConditionLeft and right subtrees of every nodehave equal heights differing by at most 1

Define: balance(x) = height(x.left) – height(x.right)

AVL property: –1 balance(x) 1, for every node x

• Ensures small depth– Will prove this by showing that an AVL tree of height

h must have a lot of (i.e. O(2h)) nodes• Easy to maintain

– Using single and double rotations

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The AVL Tree Data Structure

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Structural properties

1. Binary tree property

2. Balance property:balance of every node isbetween -1 and 1

Result:

Worst case depth isO(log n)

Ordering property– Same as for BST

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