trees main and savitch chapter 10. binary trees a binary tree has nodes, similar to nodes in a...

Trees

Main and Savitch

Chapter 10

Binary TreesBinary Trees

• A binary tree has nodes, similar to nodes in a linked list structure.

• Data of one sort or another may be stored at each node.

• But it is the connections between the nodes which characterize a binary tree.

A Binary Tree of StatesA Binary Tree of States

In this example, the data contained at each node is one of the 50 states.


Each tree has a special node called its root, usually drawn at the top.


Each tree has a special node called its root, usually drawn at the top. The example tree

has Washingtonas its root.

The example treehas Washington

as its root.


Each node is permitted to have two links to other nodes, called the left child and the right child.


Children are usually drawn below a node.

The right child ofWashington is

Colorado.

The right child ofWashington is

Colorado.

The left child ofWashington is

Arkansas.

The left child ofWashington is

Arkansas.


Some nodes have only one child.

Arkansas has aleft child, but no

right child.

Arkansas has aleft child, but no

right child.

A QuizA Quiz


Which node hasonly a right child?

Which node hasonly a right child?

A QuizA Quiz


Florida hasonly a right child.

Florida hasonly a right child.


A node with no children is called a leaf.


Each node is called the parent of its children.

Washington is theparent of Arkansas

and Colorado.

Washington is theparent of Arkansas

and Colorado.


Two rules about parents:

The root has no parent.

Every other node has exactly one parent.


Two nodes with the same parent are called siblings.

Arkansasand Coloradoare siblings.

Arkansasand Coloradoare siblings.

Complete Binary TreesComplete Binary Trees

A complete binary tree is a special kind of binary tree which will be useful to us.


A complete binary tree is a special kind of binary tree which will be useful to us.

When a completebinary tree is built,

its first node must bethe root.

When a completebinary tree is built,

its first node must bethe root.


The second node of a complete binary tree is always the left child of the root...


The second node of a complete binary tree is always the left child of the root...

... and the third node is always the right child of the root.


The next nodes must always fill the next level from left to right.

Is This Complete?Is This Complete?

Is This Complete?Is This Complete?

Yes!

It is called the empty tree, and it has no nodes, not even a root.

Implementing a Complete Binary Tree


• We will store the data from the nodes in a partially-filled array.

An array of dataWe don't care what's inthis part of the array.

An integer to keeptrack of how many nodes are in the tree

3



• We will store the date from the nodes in a partially-filled array.

An array of dataWe don't care what's inthis part of the array.

An integer to keeptrack of how many nodes are in the tree

3

Read Section 10.2 tosee details of how

the entries are stored.

Read Section 10.2 tosee details of how

the entries are stored.

Depth of Binary TreeDepth of Binary Tree

This example, depth = 3

Depth of Complete Binary Tree

Given a complete binary tree of N nodes, what is the depth?

D = 0N = 1 D = 1

N = 3 D = 2N = 7

D = 1N = 2 D = 2

N = 4

Depth of Complete Binary Tree

Given a complete binary tree of N nodes, what is the depth?

D = 0N = 1 D = 1

N = 3 D = 2N = 7

D = 1N = 2 D = 2

N = 4

D = floor(log N) = O(log N)

Depth of Binary Tree

Given a binary tree of N nodes, what is the maximum possible depth?

D = 0N = 1 D = 2

N = 3 D = 4N = 5

D = O(N)

• Binary trees contain nodes.• Each node may have a left child and a right child.• If you start from any node and move upward, you

will eventually reach the root.• Every node except the root has one parent. The

root has no parent.• Complete binary trees require the nodes to fill in

each level from left-to-right before starting the next level.

Summary Summary

• One of the tree applications in Chapter 10 is binary search trees.

• In Chapter 10, binary search trees are used to implement bags and sets.

• This presentation illustrates how another data type called a dictionary is implemented with binary search trees.

Binary Search TreesBinary Search Trees

The Dictionary Data TypeThe Dictionary Data Type

• A dictionary is a collection of items, similar to a bag.

• But unlike a bag, each item has a string attached to it, called the item's key.




Example: The items I am storing are records containing data about a state.




Example: The key for each record is the name of the state. Washington


• The insertion procedure for a dictionary has two parameters.

void Dictionary::insert(The key for the new item, The new item);

Washington


• When you want to retrieve an item, you specify the key...

Item Dictionary::retrieve("Washington");

Item Dictionary::retrieve("Washington");


When you want to retrieve an item, you specify the key... ... and the retrieval procedure returns the item.


• We'll look at how a binary tree can be used as the internal storage mechanism for the dictionary.

Arizona

Arkansas

Colorado

A Binary Search Tree of StatesA Binary Search Tree of States

The data in the dictionary will be stored in a binary tree, with each node containing an item and a key.

Washington

Oklahoma

Florida

Mass.

New

Ham

pshi

re

Wes

tV

irgin

ia

Colorado

Arizona

Arkansas


Washington

OklahomaColorado

Florida

Mass.

New

Ham

pshi

re

Wes

tV

irgin

ia

Storage rules:

1. Every key to the left of a node is alphabetically before the key of the node.

Arizona

Colorado

Arkansas


Storage rules:


Washington

Oklahoma

Florida

Mass.

New

Ham

pshi

re

Wes

tV

irgin

ia

Example: ' Massachusetts' and ' New Hampshire' are alphabetically before 'Oklahoma'

Arizona

Arkansas


Storage rules:


2. Every key to the right of a node is alphabetically after the key of the node.

Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

re

Arizona

Arkansas

Retrieving DataRetrieving Data

Start at the root.1. If the current node has

the key, then stop and retrieve the data.

2. If the current node's key is too large, move left and repeat 1-3.

3. If the current node's key is too small, move right and repeat 1-3.

Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

re

Arizona

Arkansas

Retrieve ' New Hampshire'Retrieve ' New Hampshire'

Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

re

Start at the root.

1. If the current node has the key, then stop and retrieve the data.



Arizona

Arkansas

Retrieve 'New Hampshire'Retrieve 'New Hampshire'

Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

re

Start at the root.

1. If the current node has the key, then stop and retrieve the data.



Retrieval: Complexity

• Given a complete tree of N items, the depth D = O(log N).

• What is the maximum number of nodes tested to retrieve an item from the binary search tree if we use a complete tree?

• What is the maximum number of nodes tested (worst case) to retrieve an item from a binary search tree that is not complete or balanced?

Arizona

Arkansas

Adding a New Item with aGiven Key

Adding a New Item with aGiven Key

1. Pretend that you are trying to find the key, but stop when there is no node to move to.

2. Add the new node at the spot where you would have moved to if there had been a node.

Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

re

Arizona

Arkansas

AddingAdding



Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

re

Iowa

Arizona

Arkansas

AddingAdding



Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

reIowa

Arizona

Arkansas

Adding Adding

Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

reIowa

Where would youadd this state?

Where would youadd this state?

Kazakhstan

Arizona

Arkansas

Adding Adding

Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

reIowa

Kazakhstan is thenew right child

of Iowa?

Kazakhstan is thenew right child

of Iowa?

Kazakhstan

Adding: Complexity

• Given a complete tree of N items, the depth D = O(log N).

• What is the maximum number of nodes tested to add an item to the binary search tree if we use a complete tree?

• What is the maximum number of nodes tested (worst case) to add an item from a binary search tree that is not complete or balanced?

Arizona

Arkansas

Removing an Item with a Given Key


1. Find the item.

2. If necessary, swap the item with one that is easier to remove.

3. Remove the item.Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

reIowa

Kazakhstan

Arizona

Arkansas

Removing 'Florida'Removing 'Florida'

1. Find the item.

Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

reIowa

Kazakhstan

Arizona

Arkansas


Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

reIowa

Kazakhstan

Florida cannot beremoved at the

moment...

Florida cannot beremoved at the

moment...

Arizona

Arkansas


Washington

OklahomaColorado

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

reIowa

Kazakhstan

... because removingFlorida would

break the tree intotwo pieces.

... because removingFlorida would

break the tree intotwo pieces.

Arizona

Arkansas


Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

reIowa

Kazakhstan

The problem ofbreaking the treehappens because

Florida has 2 children.

The problem ofbreaking the treehappens because

Florida has 2 children.

1. If necessary, do some rearranging.

Arizona

Arkansas


If necessary, do some rearranging.

Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

reIowa

Kazakhstan

For the rearranging,take the smallest itemin the right subtree...

For the rearranging,take the smallest itemin the right subtree...

Arizona

Arkansas


Washington

OklahomaColorado

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

re

Iowa

Kazakhstan

Iowa...copy that smallest

item onto the itemthat we're removing...

...copy that smallestitem onto the item

that we're removing...


Arizona

Arkansas


Washington

OklahomaColorado

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

re

Iowa

Kazakhstan

... and then removethe extra copy of the

item we copied...

... and then removethe extra copy of the

item we copied...


Arizona

Arkansas


Washington

OklahomaColorado

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

re

Iowa

Kazakhstan... and reconnect

the tree


Arizona

Arkansas


Washington

OklahomaColorado

Florida

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

re

Kazakhstan

Why did we choosethe smallest item

in the right subtree?

Why did we choosethe smallest item

in the right subtree?

Arizona

Arkansas


Washington

OklahomaColorado

Wes

tV

irgin

ia

Mass.

New

Ham

pshi

re

Iowa

Kazakhstan

Because every keymust be smaller than

the keys in itsright subtree



1. Find the item.

2. If the item has a right child, rearrange the tree:1. Find smallest item in the right subtree

2. Copy that smallest item onto the one that you want to remove

3. Remove the extra copy of the smallest item (making sure that you keep the tree connected)

3. else just remove the item.

• Binary search trees are a good implementation of data types such as sets, bags, and dictionaries.

• Searching for an item is generally quick since you move from the root to the item, without looking at many other items.

• Adding and deleting items is also quick.• But as you'll see later, it is possible for the

quickness to fail in some cases -- can you see why?

Summary Summary

trees main and savitch chapter 10. binary trees a binary tree has nodes, similar to nodes in a...

Documents

complete binary trees

binary tree of states

special kind of binary

right child of washington

example tree

left child of washington

special node

parent of arkansas