1. Sorted Binary Trees

2. Non-sorted binary trees
There are many uses for binary trees = x + 1
The expression x = x + 1
Mary
Bob
Ann
Jack
Jill
Tom
Tina
A genealogy: left=father, right=mother
Many—in fact, most—of these uses do not involve the idea of a “sorted” binary tree

3. Sorted binary trees
We will call a binary tree sorted if, for every node in the binary tree,
Every value in the left subtree of this node (if any) is less than the value in this node
Every value in the right subtree of this node (if any) is greater than or equal to the value in this node
It doesn’t really matter which side “equal” values go on, so long as we are consistent 15 12 20 16 25 4 8 21 6 9

4. Inserting nodes Inserting a new node is easy
If the value is smaller than this node, go left
If the value is greater or equal, go right
Add the node as a leaf
Example: add a node containing 23 to this binary tree 15 12 20 16 25 4 8 21 6 9 23 15 12 20 16 25 4 8 21 6 9

5. Deleting nodes
To delete a node from a sorted binary tree, leaving the binary tree sorted, there are three cases:
Deleting a leaf
Deleting a node with one child
Deleting a node with two children

6. Deleting a leaf Case 1: Deleting a leaf (a node with no children)
This is trivial--just remove it 15 12 20 16 25 4 8 21 6 9 6

7. Deleting a node with one child
Case 2: Deleting a node with just one child
This is simple--the child moves up to replace the deleted node 15 12 20 16 25 4 8 21 6 9

8. Deleting a node with one child
Case 2: Deleting a node with just one child
This is simple--the child moves up to replace the deleted node 15 12 20 16 25 8 21 6 9

9. Deleting a node with one child
Case 2: Deleting a node with just one child
This is simple--the child moves up to replace the deleted node 15 12 20 16 25 8 21 6 9

10. Deleting a node with two children
Case 3: Deleting a node with two children
Find the inorder successor
Replace the deleted node with its successor 15 12 20 16 25 4 8 21 6 9

11. Deleting a node with two children
Case 3: Deleting a node with two children
Find the inorder successor
Replace the deleted node with its successor 15 12 20 16 25 4 8 21 6 9

12. Deleting a node with two children
Case 3: Deleting a node with two children
Find the inorder successor
Replace the deleted node with its successor 12 20 16 25 4 8 21 6 9

13. Deleting a node with two children
Case 3: Deleting a node with two children
Find the inorder successor
Replace the deleted node with its successor 12 20 16 25 4 8 21 6 9
It gets more complicated if the inorder successor has a right child
It cannot have a left child

14. Finding the inorder successor
Let node be the node to be deleted
successor = node.rightChild; 20 10 40 30 90 60 70 50 80
while (successor.leftChild != null) successor = successor.leftChild;
This only works if the node to be deleted has a right child
We are only using it in the case where the node to be deleted has two children, so it’s good enough for our current purposes

15. If inorder successor has right child
Notice that the inorder successor cannot have a left child
If it did, that child would be the inorder successor
We need to replace the 40 with 50, as before 20 10 40 30 90 60 70 50 80

16. If inorder successor has right child
Notice that the inorder successor cannot have a left child
If it did, that child would be the inorder successor
We need to replace the 40 with 50, as before 20 10 30 90 60 70 50 80
But we also need to move the right child (and any subtree beneath it) up to the deleted node’s position

17. If inorder successor has right child
Notice that the inorder successor cannot have a left child
If it did, that child would be the inorder successor
We need to replace the 40 with 50, as before 20 10 30 90 60 70 50 80
But we also need to move the right child (and any subtree beneath it) up to the deleted node’s position

18. Observations
Deleting a node from the “middle” of a binary tree (as opposed to deleting a leaf) is an unusual operation
We did it in such a way as to keep a sorted binary tree in sorted order
This procedure would not make any sense for other, non-sorted binary trees
If we have a sorted binary tree, it is probably because we want to do searches on it
If so, we also want to keep the binary tree balanced
There are several complex algorithms for doing this

19. The End