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1
Trees III:
Binary Search Trees
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A forest full of trees
• The generic toolkit of functions we have seen thus far can be applied to many types of data structures derived from binary trees
• Now we will look at a specific ADT based on the generic binary tree: binary search binary search treestrees
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BST characteristics
• Based on binary trees
• Defining quality has to do with the order in which data are stored
• Commonly used in database applications where rapid retrieval of data is desired
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Binary Search Trees
• Entries in a BST must be objects to which total total order semanticsorder semantics apply -- in other words, objects for which all the binary comparison operators are defined
• Storage rules -- for every node n:– every entry in n’s left subtree is less than or equal
to the entry in n– every entry in n’s right subtree is greater than n’s
entry
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Binary Search Tree
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5 11
2 7 13
170 4
There is no specialrequirement for the treeto maintain to maintaina particular shape -- buta balancedbalanced tree (in which there are approximately the same number of nodes in each subtree) facilitates data searching
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The Dictionary Data TypeThe 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.
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The Dictionary Data TypeThe 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.
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The Dictionary Data TypeThe 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
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The Dictionary Data TypeThe Dictionary Data Type
• The insertion procedure for a dictionary has two parameters.
void Dictionary::insert(The key for the new item, The new item);
Washington
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The Dictionary Data TypeThe Dictionary Data Type
• When you want to retrieve an item, you specify the key...
Item Dictionary::retrieve("Washington");
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Item Dictionary::retrieve("Washington");
The Dictionary Data TypeThe Dictionary Data Type
When you want to retrieve an item, you specify the key... ... and the retrieval procedure returns the item.
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The Dictionary Data TypeThe Dictionary Data Type
• We'll look at how a binary tree can be used as the internal storage mechanism for the dictionary.
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Arizona
Arkansas
Colorado
A Binary Search Tree of StatesA Binary Search Tree of States
The data in the dictionary will be stored in a binary tree, with each node containing an item and a key.
Washington
Oklahoma
Florida
Mass.
New
Ham
pshi
re
Wes
tV
irgin
ia
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Colorado
Arizona
Arkansas
A Binary Search Tree of StatesA Binary Search Tree of States
Washington
OklahomaColorado
Florida
Mass.
New
Ham
pshi
re
Wes
tV
irgin
ia
Storage rules: Every key to the left
of a node is alphabetically before the key of the node.
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Arizona
Colorado
Arkansas
A Binary Search Tree of StatesA Binary Search Tree of States
Storage rules: Every key to the left of
a node is alphabetically before the key of the node.
Washington
Oklahoma
Florida
Mass.
New
Ham
pshi
re
Wes
tV
irgin
ia
Example: ' Massachusetts' and ' New Hampshire' are alphabetically before 'Oklahoma'
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Arizona
Arkansas
A Binary Search Tree of StatesA Binary Search Tree of States
Storage rules: Every key to the left of
a node is alphabetically before the key of the node.
Every key to the right of a node is alphabetically after the key of the node.
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
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Arizona
Arkansas
A Binary Search Tree of StatesA Binary Search Tree of States
Storage rules: Every key to the left of
a node is alphabetically before the key of the node.
Every key to the right of a node is alphabetically after the key of the node.
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
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Arizona
Arkansas
Retrieving DataRetrieving Data
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.
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
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Arizona
Arkansas
Retrieve ' New Hampshire'Retrieve ' New Hampshire'
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
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.
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Arizona
Arkansas
Retrieve 'New Hampshire'Retrieve 'New Hampshire'
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
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.
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Arizona
Arkansas
Retrieve 'New Hampshire'Retrieve 'New Hampshire'
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
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.
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Arizona
Arkansas
Retrieve 'New Hampshire'Retrieve 'New Hampshire'
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
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.
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Arizona
Arkansas
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.
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
Adding a New Item with aGiven Key
Adding a New Item with aGiven Key
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Arizona
Arkansas
AddingAdding
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.
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
Iowa
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Arizona
Arkansas
AddingAdding
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.
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
Iowa
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Arizona
Arkansas
AddingAdding
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.
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
Iowa
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Arizona
Arkansas
AddingAdding
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.
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
Iowa
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Arizona
Arkansas
AddingAdding
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.
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
Iowa
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Arizona
Arkansas
AddingAdding
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.
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
reIowa
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Arizona
Arkansas
Adding Adding
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
reIowa
Where would youadd this state?
Where would youadd this state?
Kazakhstan
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Arizona
Arkansas
Adding Adding
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
reIowa
Kazakhstan is thenew right child
of Iowa?
Kazakhstan is thenew right child
of Iowa?
Kazakhstan
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Arizona
Arkansas
Removing an Item with a Given Key
Removing an Item with a Given Key
Find the item. If necessary, swap the
item with one that is easier to remove.
Remove the item.Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
reIowa
Kazakhstan
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Arizona
Arkansas
Removing 'Florida'Removing 'Florida'
Find the item.
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
reIowa
Kazakhstan
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Arizona
Arkansas
Removing 'Florida'Removing 'Florida'
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
reIowa
Kazakhstan
Florida cannot beremoved at the
moment...
Florida cannot beremoved at the
moment...
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Arizona
Arkansas
Removing 'Florida'Removing 'Florida'
Washington
OklahomaColorado
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
reIowa
Kazakhstan
... because removingFlorida would
break the tree intotwo pieces.
... because removingFlorida would
break the tree intotwo pieces.
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Arizona
Arkansas
Removing 'Florida'Removing 'Florida'
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
reIowa
Kazakhstan
The problem ofbreaking the treehappens because
Florida has 2 children.
The problem ofbreaking the treehappens because
Florida has 2 children.
If necessary, do some rearranging.
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Arizona
Arkansas
Removing 'Florida'Removing 'Florida'
If necessary, do some rearranging.
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
reIowa
Kazakhstan
For the rearranging,take the smallest itemin the right subtree...
For the rearranging,take the smallest itemin the right subtree...
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Arizona
Arkansas
Removing 'Florida'Removing 'Florida'
Washington
OklahomaColorado
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
Iowa
Kazakhstan
Iowa...copy that smallest
item onto the itemthat we're removing...
...copy that smallestitem onto the item
that we're removing...
If necessary, do some rearranging.
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Arizona
Arkansas
Removing 'Florida'Removing 'Florida'
Washington
OklahomaColorado
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
Iowa
Kazakhstan
... and then removethe extra copy of the
item we copied...
... and then removethe extra copy of the
item we copied...
If necessary, do some rearranging.
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Arizona
Arkansas
Removing 'Florida'Removing 'Florida'
Washington
OklahomaColorado
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
Iowa
Kazakhstan... and reconnect
the tree
If necessary, do some rearranging.
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Arizona
Arkansas
Removing 'Florida'Removing 'Florida'
Washington
OklahomaColorado
Florida
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
Kazakhstan
Why did I choosethe smallest item
in the right subtree?
Why did I choosethe smallest item
in the right subtree?
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Arizona
Arkansas
Removing 'Florida'Removing 'Florida'
Washington
OklahomaColorado
Wes
tV
irgin
ia
Mass.
New
Ham
pshi
re
Iowa
Kazakhstan
Because every keymust be smaller than
the keys in itsright subtree
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Removing an Item with a Given Key
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.
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• 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.• But as you'll see later, it is possible for the
quickness to fail in some cases -- can you see why?
Summary Summary
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Trees III:
Binary Search Trees
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