AVL-Trees
(Adelson-Velskii & Landis, 1962)
In normal search trees, the complexity of find, insert
and delete operations in search trees is in the worst
case: (n).
Can be better! Idea: Balanced trees.
Definition: An AVL-tree is a binary search tree such that
for each sub-tree T ' = < L, x, R >
| h(L) - h(R) |  1 holds
(balanced sub-trees is a characteristic of AVL-trees).
The balance factor or height is often annotated at each
node h(.)+1.
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|Height(I) – hight(D)| < = 1
This is an AVL tree
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This is NOT an AVL tree (node * does not hold the required condition)
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Goals
1. How can the AVL-characteristics be kept when
inserting and deleting nodes?
2. We will see that for AVL-trees the complexity of the
operations is in the worst case
= O(height of the AVL-tree)
= O(log n)
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Preservation of the AVL-characteristics
After inserting and deleting nodes from a tree we must
procure that new tree preserves the characteristics
of an AVL-tree:
Re-balancing.
How ?: simple and double rotations
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Only 2 cases (an their mirrors)
• Let’s analyze the case of insertion
– The new element is inserted at the right (left) subtree of the right (left) child which was already
higher than the left (right) sub-tree by 1
– The new element is inserted at the left (right) subtree of the right (left) child which was already
higher than the left (right) sub-tree by 1
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Rotation (for the case when the right sub-tree grows
too high after an insertion)
Is transformed into
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Double rotation (for the case that the right sub-tree
grows too high after an insertion at its left sub-tree)
Double rotation
Is transformed into
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b
First rotation
a
x
W
a
c
new
y
Z
Second rotation
b
W
c
y
Z
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Re-balancing after insertion:
After an insertion the tree might be still balanced or:
theorem: After an insertion we need only one rotation
of double-rotation at the first node that got
unbalanced * in order to re-establish the balance
properties of the AVL tree.
(* : on the way from the inserted node to the root).
Because: after a rotation or double rotation the
resulting tree will have the original size of the tree!
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The same applies for deleting
• Only 2 cases (an their mirrors)
– The element is deleted at the right (left) sub-tree of
which was already smaller than the left (right) subtree by 1
– The new element is inserted at the left (right) subtree of the right (left) child which was already
higher that the left (right) sub-tree by 1
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The cases
Deleted node
1
1
1
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Re-balancing after deleting:
After deleting a node the tree might be still balanced or:
Theorem: after deleting we can restore the AVL balance
properties of the sub-tree having as root the first* node that
got unbalanced with just only one simple rotation or a double
rotation.
(* : on the way from the deleted note to the root).
However: the height of the resulting sub-tree might be
shortened by 1, this means more rotations might be
(recursively) necessary at the parent nodes, which can affect
up to the root of the entire tree.
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About Implementation



While searching for unbalanced sub-tree after an operation
it is only necessary to check the parent´s sub-tree only
when the son´s sub-tree has changed it height.
In order make the checking for unbalanced sub-trees more
efficient, it is recommended to put some more information
on the nodes, for example: the height of the sub-tree or
the balance factor (height(left sub-tree) – height(right subtree)) This information must be updated after each
operation
It is necessary to have an operation that returns the parent
of a certain node (for example, by adding a pointer to the
parent).
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Complexity analysis– worst case
Be h the height of the AVL-tree.
Searching: as in the normal binary search tree O(h).
Insert: the insertion is the same as the binary search tree (O(h))
but we must add the cost of one simple or double rotation,
which is constant : also O(h).
delete: delete as in the binary search tree(O(h)) but we must add
the cost of (possibly) one rotation at each node on the way
from the deleted node to the root, which is at most the height
of the tree: O(h).
All operations are O(h).
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Calculating the height of an AVL tree
Be N(h) the minimal number of nodes
In an AVL-tree having height h.
Principle of construction
0
N(0)=1, N(1)=2,
N(h) = 1 + N(h-1) + N(h-2) für h  2.
N(3)=4, N(4)=7
remember: Fibonacci-numbers
fibo(0)=0, fibo(1)=1,
fibo(n) = fibo(n-1) + fibo(n-2)
fib(3)=1, fib(4)=2, fib(5)=3, fib(6)=5, fib(7)=8
By calculating we can state:
N(h) = fibo(h+3) - 1
1
2 3
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Be n the number of nodes of an AVL-tree of height h. Then it
holds that:
n  N(h) ,
By making p = (1 + sqrt(5))/2 und q = (1- sqrt(5))/2
we can now write
n  fibo(h+3)-1
= ( ph+3 – qh+3 ) / sqrt(5) – 1
 ( p h+3/sqrt(5)) – 3/2,
thus
h+3+logp(1/sqrt(5))  logp(n+3/2),
thus there is a constant c with
h  logp(n) + c
= logp(2) • log2(n) + c
= 1.44… • log2(n) + c = O(log n).
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