Find 13:
6
2 9
7 13
Find
15
11
16
20
18 23

Find Min:
6
Find Min and Find Max
15
Find Max:
16
20
2 9
7 13
18 23
11

Insert 13:
5
2 9
Insert
12
15
18
19
17

Insert 13:
5
2 9
Insert
12
15
18
19
13 17

Delete 13:
5
2 12
Delete
15
10 13
6
7
16
20
18 23

Delete 13:
5
2 12
Delete
15
6
10
16
20
18
• No child (leaf)
• Deleted directly
23
7

Delete 16:
5
2 12
Delete
15
10 13
6
7
16
20
18 23

Delete 16:
5
2 12
Delete
15
6
10 13
16
20
18
• 1 child
• Splice out node
23
7

Delete 16:
5
2 12
Delete
15
10 13
6
7
18
20
23
• tree re-drawn for proper level layout

Delete 5:
5
2 12
Delete
15
10 13
6
7
16
20
18 23

Delete 5:
5
2 12
Delete
15
6
10
7
13
16
20
18
• 2 children
• Splice out min node of right sub-tree
23

Delete 5:
5
2 12
Delete
15
6 7
10 13
16
20
18 23
• Orphaned node’s key needs to replace key in node to be deleted

Delete 5:
6
2 12
Delete
15
10 13
7
16
20
18 23

The Effect of the Input Sequence on the Shape of BSTs
• The order in which keys arrive and are inserted impacts the shape of the resulting tree
• The shape of the tree determines performance
• All input sequences need to be accommodated in a well behaving BST data structure
• Deletions also affect tree shape (performance)

Unbalanced BST
Insert the numbers 1, 2, 3, 4, 5, 6, 7 into empty BST:
1
2
2
3
4
5
6
• Input sequence degenerated the tree into a linked list (linear) data structure.
7

Unbalanced BST
Insert the numbers 7, 6, 5, 4, 3, 2, 1 into empty BST:
7
6
5
4
3
2
1
• Tree is linear again.

Unbalanced BST
Insert the numbers 4, 3, 5, 2, 6, 1, 7 into empty BST:
4
5 3
6
2
7
1
• Tree is still very unbalanced (near-linear).

Unbalanced BST
Insert the numbers 1, 3, 2, 5, 4, 7, 6, 8 into empty BST:
1
3
2
5
4 7
6
8
• Tree is still very unbalanced (height is linear).

Balanced BST
Insert the numbers 4, 2, 6, 1, 3, 5, 7 into empty BST:
4
2 6
1
3 5
7
• Tree is “dense” with a low height.