trees1 trees and codes a bc de 000100111011 abcde

Trees 1

Trees and Codes

a

b c

d e00 010 011 10 11

a b c d e

Trees 2

Binary Trees and Prefix Codes

Imagine that you have an alphabet of symbols e.g. A, B, …, Z, a, b, …, z, ‘ ’, ‘.’

We wish to represent a string of these symbols as a string of bits e.g. “This is a string of characters”

becomes “01100111000101010111001”

Trees 3

Method 1: Use a fixed number for each symbol

Map A -> 1, B -> 2, …“This is a string of characters” becomes a string of numbers “20,34,35,…”

I required commas to separate the characters! Use a fixed width, padded with leading 0’s

instead “020034035…”

Trees 4

Method 1: Fixed width

How wide do my characters need to be Using a string of bits How many bits are needed to represent the

largest character?

ceil(log2(n)) bits

Use that many bits for each characterThis is the system used within the computer with 8 bits for each ASCII code

Trees 5

Method II: Prefix codesFor each symbol, we’ll use a code with a special property No code is the prefix of any other code

How does this work?Decoding: We read in the codes one bit at a time When we have a code we recognise, it

must be the end of a symbol It cannot be part of a longer symbol

because no code is the prefix of another code

Trees 6

Method II: Prefix codes

Example A:00, B:010, C:011, D:10, E:11

ADBECABADE00100101101100010001011

A D B E C A B A D E

The string decodes

Trees 7

Making a Prefix CodeWe want the code to be efficient No strings longer than necessary No wasted strings

A code is a set of strings of binary digits, such that no string corresponding to one symbol is the prefix of a string corresponding to another symbolIn a tree, leaf nodes have no children No path from the root to a leaf is the prefix

of a path from the root to another node

Trees 8

Binary Trees and Prefix Codes

Binary trees are in one to one correspondence with Prefix Codes A:00, B:010, C:011, D:10, E:11

A

B C

D E

0

0

0

0

1

11

1

Trees 9

Prefix Trees

Binary Trees The left child corresponds to 0, the

right to 1

Each leaf contains a symbolThe code for a symbol corresponds to the path from the root to the leaf containing that symbol

Trees 10

Encoding and Decoding

Imagine the encoder and decoder running in parallel

Encoding Start from the root While you are not at the symbol’s leaf

If the symbol you wish to send is a left decendant, send 0 and move to your left child, else send 1 and move to your right child

Decoding Start from the root While you are not at a leaf

Read a bit. If it is 0 then move to your left chile, else move to your right child

Trees 11

Encoding and Decoding: ACD

A

B C

D E A

B C

D E

Encoding:ACD Decoding:

Trees 12


A

B C

D E A

B C

D E


0

Trees 13


A

B C

D E A

B C

D E


Trees 14


A

B C

D E A

B C

D E


0

Trees 15


A

B C

D E A

B C

D E

Encoding:CD Decoding:A

Trees 16


A

B C

D E A

B C

D E


Trees 17


A

B C

D E A

B C

D E


0

Trees 18


A

B C

D E A

B C

D E


Trees 19


A

B C

D E A

B C

D E


1

Trees 20


A

B C

D E A

B C

D E


Trees 21


A

B C

D E A

B C

D E


1

Trees 22


A

B C

D E A

B C

D E

Encoding:D Decoding:AC

Trees 23


A

B C

D E A

B C

D E


Trees 24


A

B C

D E A

B C

D E


1

Trees 25


A

B C

D E A

B C

D E


Trees 26


A

B C

D E A

B C

D E


1

Trees 27


A

B C

D E A

B C

D E

Encoding: Decoding:ACD

Trees 28

Back to Method I: Balanced Tree

Method I was to used fixed length code wordsEach path from the root to a leaf is the same length: a balanced treeBalanced trees are good for worst case path length. Are they good for coding? Yes, if you assume the worst case But we can normally do better…

Trees 29

Statically optimal codes

Want common symbols to have short codesThis will make uncommon symbols have longer codes In a tree with a fixed number of

leave/symbols, moving one leaf/symbol closer to the root will move others further away

Trees 30

Huffman codes

From Shannon’s information theory, The optimal static code assigns -log2(p) bits to a symbol that occurs with probability pIt is possible to make a Huffman code tree with this property Will look at this later in the course

Trees 31

Adaptive Codes

As long as the same change is made in both sending and receiving trees/codes, there is no reason why the tree/code must remain static Send a character using the initial tree Update the tree using that character

Can also be updated in the receiver as it already has the character

Send the next character

Trees 32


A

B C

D E A

B C

D E


Trees 33


A

B C

D E A

B C

D E


00

Trees 34



A

B C D E

A

B C D E

Make same change in both trees:Rotate A’s parent

Trees 35



A

B C D E

A

B C D E

Trees 36


Encoding:D Decoding:CA

A

B C D E

A

B C D E

101

Trees 37



A

B

C

D E

A

B

C

D E

Trees 38



A

B

C

D E

A

B

C

D E

Trees 39


Encoding: Decoding:CAD

A

B

C

D E

A

B

C

D E

1110

Trees 40

Splay Trees do Rotations after Every Operation

new operation: splay splaying moves a node to the root using rotations

right rotation makes the left child x of a node y into y’s parent; y becomes the right child of x

y

x

T1 T2

T3

y

x

T1

T2T3

left rotation makes the right child y of a node x

into x’s parent; x becomes the left child of y

y

x

T1 T2

T3

y

x

T1

T2T3

(structure of tree above y is not modified)

(structure of tree above x is not modified)

a right rotation about y a left rotation about x

Trees 41

Splaying:

is x the root?

stop

is x a child of the root?

right-rotate about the root

left-rotate about the root

is x the left child of the

root?

is x a left-left grandchild?

is x a left-right grandchild?

is x a right-right grandchild?

is x a right-left grandchild?

right-rotate about g, right-rotate about p

left-rotate about g, left-rotate about p

left-rotate about p, right-rotate about g

right-rotate about p, left-rotate about g

start with node x

“x is a left-left grandchild” means x is a left child of its parent, which is itself a left child of its parent

p is x’s parent; g is p’s parent

no

yes

yes

yes

yes

yes

yes

no

no

yes zig-zig

zig-zag

zig-zag

zig-zig

zigzig

Trees 42

Visualizing the Splaying Cases

zig-zag

y

x

T2 T3

T4

z

T1

y

x

T2 T3 T4

z

T1

y

x

T1 T2

T3

z

T4

zig-zig

y

z

T4T3

T2

x

T1

zig

x

w

T1 T2

T3

y

T4

y

x

T2 T3 T4

w

T1

trees1 trees and codes a bc de 000100111011 abcde

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