avl tree name :tin ho. introduction an avl tree is another balanced binary search tree. an avl tree...

AVL TREEAVL TREE

Name :TIN HOName :TIN HO

IntroductionIntroduction An An AVL treeAVL tree is another balanced binary search is another balanced binary search

tree. tree. Named after their inventors, Named after their inventors, AAdelson-delson-VVelskii elskii

and and LLandis,andis, They were the first dynamically balanced trees They were the first dynamically balanced trees

to be proposed. to be proposed. Like red-black trees, they are not perfectly Like red-black trees, they are not perfectly

balanced, but pairs of sub-trees differ in height balanced, but pairs of sub-trees differ in height by at most 1, maintaining an by at most 1, maintaining an O(O(loglogn)n) search search time.time.

Definition of an AVL treeDefinition of an AVL tree

►Addition and deletion operations also Addition and deletion operations also take take O(O(loglogn)n) time. time.

►An AVL tree is a binary search tree An AVL tree is a binary search tree which has the following properties: which has the following properties:

►The sub-trees of every node differ in The sub-trees of every node differ in height by at most one. height by at most one.

►Every sub-tree is an AVL treeEvery sub-tree is an AVL tree. .

Balance requirement for an AVL Balance requirement for an AVL tree: the left and right sub-trees tree: the left and right sub-trees

differ by at most 1 in heightdiffer by at most 1 in height  

1. The sub-trees of every node differ in height by at most one.

2. Every sub-tree is an AVL tree.

• Be careful with this definition: it permits some apparently unbalanced trees! For example, here are some trees:

• Tree

• AVL tree? • Yes• Examination shows that each left sub-tree has a height 1 greater

than each right sub-tree.

► AVL tree?AVL tree?

► NoNo► Sub-tree with root 8 has height 4 and sub-tree with root 18 has Sub-tree with root 8 has height 4 and sub-tree with root 18 has

height 2height 2

Key termsKey terms

AVL treesAVL trees

Trees which remain Trees which remain balancedbalanced - and thus - and thus guarantee guarantee O(logn)O(logn) search times - in a search times - in a dynamic environment. Or more dynamic environment. Or more importantly, since any tree can be re-importantly, since any tree can be re-balanced - but at considerable cost - can balanced - but at considerable cost - can be re-balanced in be re-balanced in O(logn)O(logn) time. time.

AVL TREE

• AVL Tree• An AVL tree is a tree which is balanced

• We earlier defined a balanced tree as a tree whose height is [log (n+1)]

• Another definition is “A tree is balanced if the number of nodes in every left sub-tree differs by at most 1 from the number of nodes in the corresponding right sub-tree. This is known as count-balanced

• One method of count-balancing a tree is to repeatedly move the root into the sub-tree with the smaller amount of nodes

• Each move is called a shift

Gary

Eric John

Jill

Carl Mary

Fred

Ann

Hank

KimDerek

Unbalanced Tree

Unbalanced Tree

This tree has 7 nodes to the left of the root and only 3 nodes to the right of the root

To count-balance we will right shift Jill into the right sub-tree

Jill’s in-order predecessor, Hank will now become the new root

Unbalanced Tree After 1 Shift Unbalanced Tree After 1 Shift RightRight

Jill

Eric John

Hank

Carl Mary

FredAnn

Gary

KimDerek

Unbalanced Tree After 1 Shift Unbalanced Tree After 1 Shift RightRight

• The tree is still unbalanced with 6 The tree is still unbalanced with 6 nodes to the left of root and 4 nodes nodes to the left of root and 4 nodes to the leftto the left

• We perform a shift right one more We perform a shift right one more time moving Hank into the right tree time moving Hank into the right tree and making his in-order predecessor and making his in-order predecessor (Gary) the new root(Gary) the new root

• Unbalanced Tree After 2 Shifts Right

The tree is now balanced 5-5. However the Root->left subtree is not balanced i.e 3-1 so to correct that we will right shift Eric

Jill

Eric John

Gary

Carl Mary

HankAnn

Fred

KimDerek

• 1 Shift Right On Root Left SubTree

Jill

Eric

John

Gary

Carl Mary

HankAnn

Fred

Kim

Derek

Now we have an AVL Tree

Count Balance Algorithm• // return tree whose root is n, as a count balanced (AVL) tree• public BinNode countbalance(BinNode n) {• if (not n empty) then• leftnum = number of nodes in left child• rightnum = number of nodes in right child• if (leftnum > rightnum) then• loop for i going from1 to (leftnum-rightnum)/2 do right-shift root• else • loop for i going from1 to (rightnum-leftnum)/2 do left-shift root• endif• n left child = countbalance(n left child)• n right child = countbalance(n right child)• return n• }