Download - AVL Tree: Balanced Binary Search Tree

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AVL Tree: Balanced Binary Search Tree

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BST PropertyAt every node X, values inleft subtree are smaller thanthe value in X, and values in right subtree are largerthan the value in X.

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BST PropertyAt every node X, values inleft subtree are smaller thanthe value in X, and values in right subtree are largerthan the value in X.

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AVL Balance PropertyAt every node X, the heightof the left subtree differs from the height of the right subtree by at most 1.

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height(X) = max(height(left(X)), height(right(X))) + 1height() = 1

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Is this is anAVL tree?

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00


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Yes,this is anAVL tree.

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Suppose we deletethe 9 node.

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AVL balancepropertyfails at6 node.

Tree Rotations to Restore Balance

j

k

X Y

Z

Suppose we have balance . . .


j

k

XY

Z

Suppose we have balance . . .

. . . but we insert here,destroying the balance.


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k

XY

Z

Do a “single rotation”


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k

XY Z



jk

XY Z



jk

X Y Z



jk

X Y Z


Done!


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k

XY

Z

In one step:


jk

X Y Z


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k

X Y

Z

Again, suppose we have balance . . .


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k

XY

Z

Again, suppose we have balance . . .

. . . but this time weinsert here.


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XY

Z

Again, we do a single rotation . . .


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X

YZ

Again, we do a single rotation . . .

. . . but it fails torestore balance


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k

XY

Z

Go back to where westarted . . .


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k

XY

Z

Go back to where westarted . . .

. . . and consider structure of Y


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X

V

Z

W

i


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k

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V

Z

W

i

Now do a “double rotation”. . .


jk

X V ZW

i Now do a “double rotation”. . .


jk

X V ZW

i Now do a “double rotation”. . .

Done!

Download - AVL Tree: Balanced Binary Search Tree

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