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[\<- 05/22](05-22.md)
---
## Just One of Many
- Today we will go over AVL tree insertion and deletion
- The best way to learn AVL trees: practice, practice, practice!
- AVL trees are just one type of balanced search tree:
- Red-Black trees (not as strict as AVL);
- Splay trees (change tree on search as well);
- 2-3 trees (2 or 3 children per node);
- B-trees (generalization of 2-3 trees);
- B+ trees (a B-tree on disk in which full data is stored only in the leaves, which are linked together as a linked list)
## Coding
- Coding a BST isn't that bad
- Coding an AVL tree is much harder
## Not Just Numbers
- Students always ask why are we inserting numbers into a BST?
- In reality the number is just a key that identifies a much larger piece of data
- The key might be your student ID number, which identifies your student record
- We can use strings (just like in a hash table) on a search tree as easily as we can use numbers
- We'll have fun with some strings today
# Examples
## Example 1
Insert 8, 6, 7, 5, 3, 0, 9
- Insert 8
- 8 is the root
- Insert 6
- 6 is less than 8, so it becomes the left child
- 6 is even, 8 is left high
- Insert 7
- 7 is less than 8, and higher than 6, so 7 becomes the right child of 6
- Now 8 is unbalanced
- Rotate 6 left: now the tree is root 8, 7 is the left child, 6 is the left child of 7
- Rotate 8 right: now 7 is the root, 6 is the left child, 8 is the right child
- Insert 5
- 5 is less than 7 and less than 6, so 5 becomes the left child of 6
- 6 and 7 are left high, the other nodes are even
- Insert 3
- 3 is less than 7, less than 6, and less than 5, so it becomes the left child of 5
- 6 and 7 are unbalanced
- Rotate 6 right: Now 7 is the root, 8 is the right child, 5 is the left child, 3 is the left child of 5, 6 is the right child of 5
- Insert 0
- 0 is less than 7, less than 5, less than 3, so it becomes the left child of 3
- Now the root is unbalanced
- Rotate 7 (the root) right: Now, 5 is the root, 3 is the left child, 0 is the left child of 3, 7 is the right child of the root, 6 is the left child of 7, 8 is the left child of 7
- Insert 9
- 9 is greater than 5, greater than 7, greater than 8, so it becomes the right child of 8.
- 5, 7, and 8 are right high
- Resulting Tree Stats:
- Height = 4
- Root = 5
- Leaves = {0, 6, 9}
- Single Rotations = 2
- Double Rotations = 1
- Possible scenarios
- Inserting 4 -> no rotation
- Inserting 10 -> single rotation
- Inserting 2 -> double rotation
## Example 2
- We have a tree:
- 5 is the root
- 3 is the left child
- 0 is the left child
- 7 is the right child
- 6 is the left child
- 8 is the right child
- 9 is the right child
- Delete 3: The resulting tree is:
- 5 is the root
- 0 is the left child
- 7 is the right subchild
- 6 is the left subchild
- 8 is the right subchild
- 9 is the right subchild
- The tree is right of right: 5 needs to be rotated left
- 7 is the root
- 5 is the left child
- 0 is the left child
- 6 is the right child
- 8 is the right child
- 9 is the right child
- Delete 9: The resulting tree is:
- 7 is the root
- 5 is the left child
- 0 is the left child
- 6 is the right child
- 8 is the right child
- The emphasis will be on insertion, but it is good to practice deletion regardless
## Example 3
- Insert: 10, 20, 30, 40, 50, 60, 70
- This is a worst case for a BST, but not for an AVL tree!
- Insert 10: The resulting tree is:
- 10 is the root
- Insert 20: The resulting tree is:
- 10 is the root
- 20 is the right child
- Insert 30: The resulting tree is:
- 10 is the root
- 20 is the right child
- 30 is the right child
- The tree is right of right: rotate 10 left:
- 20 is the root
- 10 is the left child
- 30 is the right child
- Insert 40: the resultign tree is:
- 20 is the root
- 10 is the left child
- 30 is the right child
- 40 is the right child
- Insert 50: the resultign tree is:
- 20 is the root
- 10 is the left child
- 30 is the right child
- 40 is the right child
- 50 is the right child
- The tree is right of right: rotate 30 left:
- 20 is the root
- 10 is the left child
- 40 is the right child
- 30 is the left child
- 50 is the right child
- etc.
## Example 4: With Words!
- Insert dog, cat, bat, fox, elk, gnu, ant
- Insert dog:
- "dog" is the root
- Insert cat:
- "dog" is the root
- "cat" is the left child
- Insert bat:
- "dog" is the root
- "cat" is the left child
- "bat" is the left child
- The root is left of left: rotate "dog" right:
- "cat" is the root
- "bat" is the left child
- "dog" is the right child
- Insert fox:
- "cat" is the root
- "bat" is the left child
- "dog" is the right child
- "fox" is the right child
- Insert elk:
- "cat" is the root
- "bat" is the left child
- "dog" is the right child
- "fox" is the right child
- "elk" is the left child
- "dog" is left of right
- first rotate "fox" right, then rotate "dog" left
- "cat" is the root
- "bat" is the left child
- "elk" is the right child
- "dog" is the left child
- "fox" is the right child
- Insert gnu:
- "cat" is the root
- "bat" is the left child
- "elk" is the right child
- "dog" is the left child
- "fox" is the right child
- "gnu" is the right child
- the root is right of right: rotate "cat" left:
- "elk" is the root
- "cat" is the left child
- "bat" is the left child
- "dog" is the right child
- "fox" is the right child
- "gnu" is the right child
- etc.
## Example 5: The Mammoth Example
- Insert doe, emu, gar, fly, boa, cow, sow, jay, hog, yak, lar, nit, ass, owl, pug, rat, man
- Try this on your own. I will only write out one guide to constructing the tree (as opposed to rewriting each time a new entry is added
- "jay" is the root
- emu" is the left child
- "cow" is the left child
- "boa" is the left child
- "ass" is the left child
- "doe" is the right child
- "gar" is the right child
- "fly" is the left child
- "hog" is the right child
- "sow" is the right child
- "lar" is the left child
- "nit" is the right child
- "yak" is the right child
- This is the AVL tree with most of the entries. Try adding the rest yourself
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