1. AVL Tree
Chapter 6 (cont’)

2. Search Trees Two standard search trees:
Binary Search Trees (non-balanced)
All items in left sub-tree are less than root
All items in right sub-tree are greater than or equal to the root
Each sub-tree is a binary search tree
AVL trees (balanced)

3. 1- Binary Search Trees(review)

4. BST Traversals A preorder traversal of the BST produces:
A postorder traversal of the BST produces:
An inorder traversal of the BST produces:
(a sorted list!)

5. BST Search Algorithms
Find Smallest Node: recursively follow the left branch until reaching a leaf
Find Largest Node: recursively follow the right branch until reaching a leaf
BST Search: recursively search for key node in a sub-tree (similar to the binary search)

6. Algorithm : Find Smallest
algorithm findSmallestBST (val root <pointer>)
if (root->left null)
return (root)
end if
return findSmallestBST (root->left)
end findSmallestBST

7. Algorithm : Find Largest
algorithm findLargestBST(val root <pointer>)
if (root->right null)
return (root)
end if
return findLargestBST(root->right)
end findLargestBST

8. Algorithm : BST Search
algorithm searchBST (ref root <pointer>, val argument <key>)
if (root is null)
return null
end if
if (argument < root->key)
return searchBST (root->left, argument)
elseif (argument > root->key)
return searchBST (root->right, argument)
else
return root
end searchBST

9. Algorithm : Recursively add node to BST
algorithm addBST (ref root <pointer>, val new <pointer>)
if (root is null)
root = new
else
Locate null sub-tree for insertion
if (new->key < root->key)
addBst (root->left, new)
addBst (root->right, new)
end if
return
end addBST

10. Insert

11. Insert

13. Delete

14. Delete

15. 2- AVL Trees Invented by Adelson-Velskii and Landis
Height-balanced binary search tree where the heights of sub-trees differ by no more than 1:
| HL – HR | <= 1
Search effort for an AVL tree is O(log2n)
Each node has a balance factor
Balance factors may be:
Left High (LH) = +1 (left sub-tree higher than right sub-tree)
Even High (EH) = 0 (left and right sub-trees same height)
Right High (RH) = -1 (right sub-tree higher than left sub-tree)

16. Figure 8-12

17. Figure 8-13 An AVL tree

18. Balancing Trees
Insertions and deletions potentially cause a tree to be imbalanced
When a tree is detected as unbalanced, nodes are balanced by rotating nodes to the left or right
Four imbalance cases:
Left of left: the sub-tree of a left high tree has become left high
Right of right: the sub-tree of a right high tree has become right high
Right of left: the sub-tree of a left high tree has become right high
Left of right: the sub-tree of a right high tree has become left high
Each imbalance case has simple and complex rotation cases

21. Case 1: Left of Left
The sub-tree of a left high tree has become left high
Simple right rotation: Rotate the out of balance node (the root) to the right
Complex right rotation: Rotate root to the right, so the old root is the right sub-tree of the new root; the new root's right sub-tree is connected to the old root's left sub-tree

23. Case 2: Right of Right
Mirror of Case 1: The sub-tree of a right high tree has become right high
Simple left rotation: Rotate the out of balance node (the root) to the left
Complex left rotation: Rotate root to the left, so the old root is the left sub-tree of the new root; the new root's left sub-tree is connected to the old root's right sub-tree

25. Case 3: Right of Left
The sub-tree of a left high tree has become right high
Simple double rotation right: Rotate left sub-tree to the left; rotate root to the right, so the old root's left node is the new root
Complex double rotation right: Rotate the right-high left sub-tree to the left; rotate the left-high left sub-tree to the right

27. Case 4: Left of Right
Mirror of Case 3: The sub-tree of a right high tree has become left high
Simple double rotation right: Rotate right sub-tree to the right; rotate root to the left, so the old root's right node is the new root
Complex double rotation right: Rotate the left-high right sub-tree to the right; rotate the right-high right sub-tree to the left

29. AVL Node Structure
Node key <keyType> data <dataType> Left <pointer to Node> right <pointer to Node> bal <LH, EH, RH> // Balance factor End Node