Copyright 2004-2006 Curt Hill Balance in Binary Trees Impact on Performance

Copyright 2004-2006 Curt Hill

Balance in Binary Trees

Impact on Performance


Tree Shape and Performance

• A tree that is balanced has excellent performance

• O(log2N) for:– Searches– Insertions– Deletions

• Only a hash table can beat this performance– But it has its own issues


What is balance?

• The notion is that the two sub-trees are of about the same size

• Thus a search eliminates half the tree in each examination

• Perfect balance:– For each node in the tree, the size of

the two sub-trees are off by at most one


Probabilities• What is the likelihood that a randomly

built tree will have good performance characteristics?

• This is a difficult question• The shape of a tree is dependent on the

entry order of the nodes to be inserted• Example:

– Consider the integers 1-7 as the items to put in a tree

– There are 7! = 5040 ways to order their input

• 7 ways to choose first• 6 ways to choose second• etc.


What do we want?

4

6

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A search must look at no more than 3 nodes


Example Continued• There are two really bad ways to

choose the tree:– In ascending order or descending

order– There are only two of these but there

are several others that are just as bad– Consider 1 7 6 5 4 3 2 or

• 1 2 3 7 6 5 4

• Bad in this case means that every node has zero or one descendents


What do we not want?1

A search must look at no more than 7 nodes

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3

4

2

6

7

1

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3

7

65

4

Arrival in ascending order

Equally bad


Negative Combinatorics• There are two ways to choose the

first item– Each subsequent item provides two

ways:– The next item in ascending order– The last item– Therefore 2 * 2 * 2 * 2 * 2 * 2 * 1– Looks like 64 ways to choose a list– This is 1.27% chance of a list

• A search would look at no more than 7 nodes


Positive Combinatorics• There is only one way to choose

the root, it must be the 4• There are two ways to choose the

second: 2 or 6• There are three ways to choose

the third– If 2 was picked the 6 or any

descendent of 2– If 6 was picked the 2 or any

descendent of 6• It gets exciting after that


Positive Combinatorics• Sub-cases need to be examined of

the three last choices• These do not work well in this kind

of presentation• I believe that there are 80 out of

5040 (1.5%) permutations that yield a perfectly balance tree

• However, most possibilities fall somewhere in between maximum pathes of 7 and 3


Summary• The worst case is a linked list which is

bad– The worst case is not very likely

• The best case is perfectly balanced– The best case is more likely, but still unlikely

• Empirical studies indicate that the average path length of a unbalanced tree to be only 39% longer than a perfectly balanced tree

• Balancing is hard and slows insertions and deletions


When to Balance

• In most cases an unbalanced tree will perform quite adequately

• If the application fulfills the following two criteria then balancing could be considered– The data is large and the search

performance impacts the program– The number of searches is large

compared to insertion and deletion


Perfectly balanced trees

• Definition:– For each node the number of nodes of

the left and right sub-trees differ by only 1

• Balancing a tree is a recursive process that involves nodes from the leaves to root

• It is usually the case that control information is placed in node that measures the balance


Balance Again

• Balancing occurs in insertion and deletion, but not searches

• It is somewhat intricate so perfect balance is seldom used

• The ratio of searches to inserts and deletes must be very high

• Is there another definition of balance that gives good performance with less rebalancing


Height Balanced

• Also known as AVL balance– Adelson, Velski and Landis – Developed it and proved its desirability

• Definition:– The tree is balanced if for each node

the heights of the two sub-trees differ at most by one

• It is the height of the tree that determines the worst case search


Digression on Search

• Consider searching an array• On average the search requires

½N comparisons• The worst case is N searchs to find

last one or to show not found• The average and worst case are

quite different • This is not the case for trees


Searching Trees4

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More than half the nodes are leaves at maximum depth. Worst case is three probes, but average case is only slightly less than three probes.


AVL Trees Again• Adelson, Velski and Landis proved:

– Worst case of an AVL tree is only 45% worse than perfectly balanced

– Average case: Insignificantly different than perfectly balanced

• Every perfectly balanced is also AVL balanced

• Far fewer rebalance, thus cheaper to construct– For the most part rebalancing occurs

when really needed


Construction

• Consider the construction of the following tree

• Four types of rebalancing operation– RR single– LL single– LR double– RL double

• Add: 4 5 7 2 1 3 6


After 2 inserts

4

5

Still perfectly balanced


Insert 7

4

5

7

Neither perfect nor AVL, rebalance is needed


Rotate Right

4

5

7

Rebalance is needed – RR Single


After Rotate

5

7

After rebalance

4


Insert 2

5

7

No problem

4

2


Insert 1

5

7

Unbalanced in other way – Do a LL single

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2

1


Rebalance

5

7

Rebalance complete – not perfect but AVL

2

1 4


Insert 3

5

7

A rebalance is again needed, but different

2

1 4

3


After Rotatation

4

5

This requires LR double

2

1 37


Insert 6

4

5

This requires RL double

2

1 37

6


Rotate 6-7

4

5

This requires RL double

2

1 36

7


Rotate 5-6-7

4

6

Now complete

2

1 375


The problem of balancing• To implement requires extra stuff in

the nodes• Measures the height of the

descendents• Even with an AVL tree there is

substantial work to be done at insertion and deletion time

• Thus the search to insert and delete ratio needs to be high– Just not as high as perfect balance


Synonyms

• Another name for an AVL trees is Fibonacci tree

• The fact that heights may disagree by one leads to as strangely asymmetric tree


Is this balanced?

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Documents

Copyright 2004-2006 Curt Hill Balance in Binary Trees Impact on Performance