Binary Search Tree
A binary search tree (BST) is a binary tree that,
for each node n:
The elements in the left subtree are less than
the element in n
The elements in the right subtree are greater
than or equal to the element in n
This definition extends the basic binary tree to
allow insertions and deletions
The elements must have an inherent ordering (be
Comparable)
Binary Search Tree
45
12
7
69
15
14
51
81
42
33
25
42
38
70
44
Adding an Element
To add an element:
Follow the appropriate path until it ends and
add the new node at that point
Add the following elements to an initially empty
tree: 38 60 87 24 56 19 45 59 22
38
24
60
19
56
22
45
87
59
Finding an Element
To find an element:
Starting at the root, compare elements,
then move into the left or right subtree as
appropriate
Eventually the target will be found or the
end of the path will be reached
For a generally balanced tree, about half of the
remaining elements are eliminated with each
comparison
Degenerate Trees
A degenerate tree is one which is highly
unbalanced
23
28
35
33
40
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57
Defeats the purpose of efficient search
Removing an Element
After removing an element, the result must still
be a valid BST
Removing a node may disconnect parts of the
tree
Three distinct situations:
1. The node to remove is a leaf
2. The node to remove has one child
3. The node to remove has two children
The first two of these are easy to deal with
Removing an Element
51
Situation 3
28
72
25
36
18
11
64
33
24
30
49
39
37
Situation 1
68
44
34
41
77
76
85
74
50
Situation 2
Removing an Element
A leaf can simply be deleted
A node with one child is simply bypassed,
"pulling up" its one subtree
A node with two children can be replaced with
its inorder successor or its inorder predecessor
Inorder successor - greatest element in the
left subtree
Inorder predecessor - least element in the
right subtree
Either will have at most one child
Efficiency
The cost of each operation is a function of the
number of nodes it must access
For a balanced tree, these operations are
logarithmic, or O(log N)
The worst case (a fully degenerate tree) is O(N)
If all permutations are equally likely, the
average case is O(N log N)
Various strategies exist for keeping a BST
balanced