Binary Search Trees
One of the tree applications in
Chapter 10 is binary search trees.
 In Chapter 10, binary search trees
are used to implement bags.
 This presentation illustrates how
another data type called a
dictionary is implemented with
binary search trees.

Binary Search Trees (BST)
A
data structure for efficient searching, insertion
and deletion
 Binary search tree property
 For
every node x
 All the keys in its left
subtree are smaller than
the key value in x
 All the keys in its right
subtree are larger than the
key value in x
Binary Search Trees
A binary search tree
Not a binary search tree
Binary Search Trees
The same set of keys may have different BSTs
 Average
depth of a BST is O(log n)
 Maximum depth of a BST is O(n)
The Dictionary Data Type
A dictionary is a collection
of items, similar to a bag.
 But unlike a bag, each item
has a string attached to it,
called the item's key.

The Dictionary Data Type
A dictionary is a collection
of items, similar to a bag.
 But unlike a bag, each item
has a string attached to it,
called the item's key.

Example:
The items I am
storing are records
containing data
about a state.
The Dictionary Data Type
A dictionary is a collection
of items, similar to a bag.
 But unlike a bag, each item
has a string attached to it,
called the item's key.

Example:
The key for each
record is the name
of the state.
Washington
The Dictionary Data Type
void dictionary::insert(The key for the new item, The new item);

The insertion procedure for a
dictionary has two
parameters.
The Dictionary Data Type

When you want to retrieve an
item, you specify the key...
Item dictionary::retrieve("Washington");
The Dictionary Data Type

When you want to retrieve an
item, you specify the key...
... and the retrieval procedure
returns the item.
Item dictionary::retrieve("Washington");
The Dictionary Data Type

We'll look at how a binary
tree can be used as the
internal storage mechanism
for the dictionary.
A Binary Search Tree of States
Oklahoma
Colorado
Mass.
Arizona
Arkansas
West
Virginia
Washington
New
Hampshire
The data in the
dictionary will
be stored in a
binary tree,
with each node
containing an
item and a key.
A Binary Search Tree of States
Oklahoma
Colorado
Mass.
Arizona
Arkansas
West
Virginia
Washington
New
Hampshire
Storage rules:
 Every key to the left
of a node is
alphabetically before
the key of the node.
A Binary Search Tree of States
Mass.
Washington
Arkansas
West
Virginia
Example:
“Massachusetts” and
“ New Hampshire”
are alphabetically
before “Oklahoma”
Oklahoma
New
Hampshire
Storage rules:
 Every key to the left of
Colorado
a node is
alphabetically before
the key of the node. Arizona
A Binary Search Tree of States
Oklahoma
Mass.
West
Virginia
Washington
New
Hampshire
Storage rules:
 Every key to the left of
Colorado
a node is
alphabetically before
the key of the node. Arizona
 Every key to the
right of a node is
alphabetically after
Arkansas
the key of the node.
A Binary Search Tree of States
Oklahoma
Mass.
West
Virginia
Washington
New
Hampshire
Storage rules:
 Every key to the left of
Colorado
a node is
alphabetically before
the key of the node. Arizona
 Every key to the
right of a node is
alphabetically after
Arkansas
the key of the node.
Searching BST
 If
we are searching for 15, then we are done.
 If we are searching for a key < 15, then we
should search in the left subtree.
 If we are searching for a key > 15, then we
should search in the right subtree.
Searching (Find)
 Find
X: return a pointer to the node that has
key X, or NULL if there is no such node
 Time
complexity: O(depth of the tree)
Retrieving Data
Oklahoma
Colorado
Mass.
Arizona
Arkansas
West
Virginia
Washington
New
Hampshire
Start at the root.
 If the current node has
the key, then stop and
retrieve the data.
 If the current node's
key is too large, move
left and repeat 1-3.
 If the current node's
key is too small, move
right and repeat 1-3.
Retrieve " New Hampshire"
Oklahoma
Colorado
Mass.
Arizona
Arkansas
West
Virginia
Washington
New
Hampshire
Start at the root.
 If the current node has
the key, then stop and
retrieve the data.
 If the current node's
key is too large, move
left and repeat 1-3.
 If the current node's
key is too small, move
right and repeat 1-3.
Retrieve "New Hampshire"
Oklahoma
Colorado
Mass.
Arizona
Arkansas
West
Virginia
Washington
New
Hampshire
Start at the root.
 If the current node has
the key, then stop and
retrieve the data.
 If the current node's
key is too large, move
left and repeat 1-3.
 If the current node's
key is too small, move
right and repeat 1-3.
Retrieve "New Hampshire"
Oklahoma
Colorado
Mass.
Arizona
Arkansas
West
Virginia
Washington
New
Hampshire
Start at the root.
 If the current node has
the key, then stop and
retrieve the data.
 If the current node's
key is too large, move
left and repeat 1-3.
 If the current node's
key is too small, move
right and repeat 1-3.
Retrieve "New Hampshire"
Oklahoma
Colorado
Mass.
Arizona
Arkansas
West
Virginia
Washington
New
Hampshire
Start at the root.
 If the current node has
the key, then stop and
retrieve the data.
 If the current node's
key is too large, move
left and repeat 1-3.
 If the current node's
key is too small, move
right and repeat 1-3.
Inorder Traversal of BST
 Inorder
traversal of BST prints out all the
keys in sorted order
Inorder: 2, 3, 4, 6, 7, 9, 13, 15, 17, 18, 20
findMin/ findMax
Goal: return the node containing the smallest
(largest) key in the tree
 Algorithm: Start at the root and go left (right) as long
as there is a left (right) child. The stopping point is
the smallest (largest) element


Time complexity = O(depth of the tree)
Insertion
 Proceed
down the tree as you would with a find
 If x is found, do nothing (or update something)
 Otherwise, insert x at the last spot on the path
traversed
 Time complexity =
O(depth of the tree)
Deletion
 When
we delete a node, we need to consider
how we take care of the children of the
deleted node.
 This
has to be done such that the property of the
search tree is maintained.
Deletion under Different Cases
 Case
1: the node is a leaf
 Delete
 Case
it immediately
2: the node has one child
 Adjust
node
a pointer from the parent to bypass that
Adding a New Item with a
Given Key
Oklahoma
Colorado
Mass.
Arizona
Washington
Arkansas
West
Virginia

Pretend that you are
trying to find the key,
but stop when there is
no node to move to.
Add the new node at
the spot where you
would have moved to
if there had been a
node.
New
Hampshire


Pretend that you are
trying to find the key,
but stop when there is
no node to move to.
Add the new node at
the spot where you
would have moved to
if there had been a
node.
Oklahoma
Colorado
Mass.
Arizona
Washington
Arkansas
West
Virginia

New
Hampshire
Adding
Iowa

Pretend that you are
trying to find the key,
but stop when there is
no node to move to.
Add the new node at
the spot where you
would have moved to
if there had been a
node.
Oklahoma
Colorado
Mass.
Arizona
Washington
Arkansas
West
Virginia

New
Hampshire
Adding
Iowa

Pretend that you are
trying to find the key,
but stop when there is
no node to move to.
Add the new node at
the spot where you
would have moved to
if there had been a
node.
Oklahoma
Colorado
Mass.
Arizona
Washington
Arkansas
West
Virginia

New
Hampshire
Adding
Iowa

Pretend that you are
trying to find the key,
but stop when there is
no node to move to.
Add the new node at
the spot where you
would have moved to
if there had been a
node.
Oklahoma
Colorado
Mass.
Arizona
Washington
Arkansas
West
Virginia

New
Hampshire
Adding
Iowa

Pretend that you are
trying to find the key,
but stop when there is
no node to move to.
Add the new node at
the spot where you
would have moved to
if there had been a
node.
Oklahoma
Colorado
Mass.
Arizona
Washington
Arkansas
West
Virginia

New
Hampshire
Adding
Iowa
Adding
Oklahoma
Colorado
Mass.
Arizona
Washington
Iowa
Arkansas
West
Virginia

Pretend that you are
trying to find the key,
but stop when there is
no node to move to.
Add the new node at
the spot where you
would have moved to
if there had been a
node.
New
Hampshire

Adding
Oklahoma
Colorado
Mass.
Arizona
Iowa
Arkansas
West
Virginia
Washington
New
Hampshire
Where would you
add this state?
Adding
Oklahoma
Colorado
Mass.
Arizona
Iowa
Arkansas
West
Virginia
Washington
New
Hampshire
Kazakhstan is the
new right child
of Iowa?
Deletion
 When
we delete a node, we need to consider
how we take care of the children of the
deleted node.
 This
has to be done such that the property of the
search tree is maintained.
Deletion under Different Cases
 Case
1: the node is a leaf
 Delete
 Case
it immediately
2: the node has one child
 Adjust
node
a pointer from the parent to bypass that
Deletion Case 3

Case 3: the node has 2 children
Replace the key of that node with the minimum element at the
right subtree
 Delete that minimum element

Has either no child or only right child because if it has a left child, that
left child would be smaller and would have been chosen. So invoke case
1 or 2.
Time complexity = O(depth of the tree)


Removing an Item with a
Given Key

Oklahoma
Colorado
Mass.
Arizona
Washington
Iowa
Arkansas
West
Virginia

Find the item.
If necessary, swap the
item with one that is
easier to remove.
Remove the item.
New
Hampshire

Removing "Florida"
Find the item.
Oklahoma
Colorado
Mass.
Arizona
Iowa
Arkansas
West
Virginia
Washington
New
Hampshire

Removing "Florida"
Oklahoma
Colorado
Mass.
Arizona
Iowa
Arkansas
West
Virginia
Florida cannot be
removed at the
moment...
New
Hampshire
Washington
Removing "Florida"
Oklahoma
Colorado
Mass.
Iowa
Arkansas
West
Virginia
Washington
New
Hampshire
... because removing
Florida would
break the tree into
two pieces.
Arizona
Removing "Florida"

If necessary, do some
rearranging.
Oklahoma
Colorado
Mass.
Iowa
Arkansas
West
Virginia
Washington
New
Hampshire
The problem of
breaking the tree
happens because
Florida has 2 children.
Arizona
Removing "Florida"
If necessary, do some
rearranging.
Oklahoma
Colorado
Mass.
Arizona
For the rearranging,
take the smallest item
in the right subtree...
Iowa
Arkansas
West
Virginia
Washington
New
Hampshire

Removing "Florida"
Iowa
If necessary, do some
rearranging.
Oklahoma
Colorado
Mass.
Arizona
...copy that smallest
item onto the item
that we're removing...
Iowa
Arkansas
West
Virginia
Washington
New
Hampshire

Removing "Florida"
Iowa
If necessary, do some
rearranging.
Oklahoma
Colorado
Mass.
Arizona
... and then remove
the extra copy of the
item we copied...
Arkansas
West
Virginia
Washington
New
Hampshire

Removing "Florida"
Iowa
If necessary, do some
rearranging.
Oklahoma
Colorado
Mass.
Arizona
... and reconnect
the tree
Arkansas
West
Virginia
Washington
New
Hampshire

Removing "Florida"
Oklahoma
Colorado
Mass.
Arkansas
West
Virginia
Washington
New
Hampshire
Why did I choose
the smallest item
in the right subtree?
Arizona
Removing "Florida"
Iowa
Oklahoma
Colorado
Mass.
Arizona
Arkansas
West
Virginia
Washington
New
Hampshire
Because every key
must be smaller than
the keys in its
right subtree
Removing an Item with a
Given Key


Find the item.
If the item has a right child, rearrange the tree:
Find smallest item in the right subtree
 Copy that smallest item onto the one that you
want to remove
 Remove the extra copy of the smallest item
(making sure that you keep the tree connected)

else just remove the item.
Summary
Binary search trees are a good implementation of
data types such as sets, bags, and dictionaries.
 Searching for an item is generally quick since you
move from the root to the item, without looking at
many other items.
 Adding and deleting items is also quick.
