Recursion and Binary Trees

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Recursion and Binary Trees

• MINS 116
• Spring 2000
• Data Structures

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Topics

• Recursion
• Binary Search - Review
• Binary Trees
• Balancing Trees

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Recursion

• Recursive of, relating to, or constituting a
  procedure that can repeat itself indefinitely or
  until a specified condition is met.

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Recursion

• Recursion is when a function calls itself from
  within itself.
• E.g.
• public int countDown( int count )
• if ( count 0 )
• count--
• retVal countDown( count
  )
• return retVal
• ...

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Recursion

• All recursive methods require
• Argument (or arguments) passed into the method.
• Return value resulting from the recursive call.
• Logical test to terminate the recursive call.

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Recursion - Stacks

• To understand what is happening with recursion,
  we must understand a data structure called
  stacks.
• A Stack is known as a Last In - First Out data
  structure (LIFO).

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Recursion - Stacks

• How a Stack works - A stack is just like a stack
  of books, papers, lunch trays, etc.
• When you add something to a stack, it gets placed
  on the top. This is called a push event.

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Recursion - Stacks

• Push Item 3

After
Before
Item 3 is put on top of the stack.
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Recursion - Stacks

• To remove something from a stack, we can only
  take off the top. This operation is called a
  pop.

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Recursion - Stacks

• Pop Item 3

After
Before
Item 3 is removed from the top of the stack.
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Recursion - Stacks

• How do stacks relate to recursion?
• The Java virtual machine uses a stack called a
  call stack.
• Every time a method call is made, the return
  address is pushed onto the stack.
• When a return is made from a method call, the
  return address is popped off of the stack.
• The stack keeps track of where the call came
  from allowing the ability to get back to the
  original caller.

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Recursion - Stacks

• You can simulate the call stack by using a stack
  structure in your programs. For instance
• In the binary tree, you can push the last node
  you have visited, giving the ability to retrace
  your steps in the tree traversal.

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Binary Search - Review

• What types of searches are there?
• What is the efficiency of these searches?

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Binary Search - Review

• I am thinking of a number between 1 and 100.
• I will tell you if you need to guess higher or
  lower.
• Guess the number.

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Binary Search - Review

• What condition must be true to use a binary
  search?
• How many comparisons are required to find an item
  in a binary search?

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Binary Trees

• A binary tree is a data structure used to store
  information in a sorted order.
• A node in a binary tree has reference to an item
  which is greater and an item which is less.
• As in linked lists, a node is merely a place
  holder for the information being contained in the
  tree.

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Binary Trees
Level Items
1 1
2 3
3 7
4 15
5 31
In general, the number of items you can store is
one less than the number 2 raised to the Nth
power where N is the number of levels. 2 - 1
N
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Binary Trees - Building

• Building of a binary tree requires the insertion
  of a node based on the information contained
  within the node.
• Example insert the integers 16, 8, 24, 20, 3 and
  25 into a binary tree structure.

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Binary Trees - Building

• 16

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Binary Trees - Building

• 16, 8

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Binary Trees - Building

• 16, 8, 24

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Binary Trees - Building

• 16, 8, 24, 20

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Binary Trees - Building

• 16, 8, 24, 20, 3

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Binary Trees - Building

• 16, 8, 24, 20, 3, 25

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Binary Trees - Traversal

• Definitions
• V - Visit the node. This is when the information
  referenced in the node is accessed.
• L - Traverse the node to the left.
• R - Traverse the node to the right.
• These rules apply recursively to each node of the
  tree.

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Binary Trees - Traversal

• Preorder V L R
• What is the preorder traversal of

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Binary Trees - Traversal

• Inorder L V R
• What is the inorder traversal of

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Binary Trees - Traversal

• Postorder L R V
• What is the postorder traversal of

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Binary Trees - Traversal

• Questions
• How can a stack be used to help with binary tree
  traversal?
• How can recursion be used in binary tree
  traversal?
• How are recursion and the use of a stack
  different?

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Binary Trees

• Exercise
• Build a binary tree out of the number sequence
  1, 3, 5, 7, 13, 24, 31
• What do you notice about the tree that was
  constructed?

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Binary Trees

• Exercise (contd)
• Does the above binary tree have any advantage
  over a singly linked list?
• What feature of the data sequence caused the
  special tree configuration?
• How might you balance this tree?

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Binary Trees - Deleting a node

• How would you go about deleting node 24 from the
  binary tree?

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Binary Trees - Deleting a node

• Cases to consider
• Deletion of a leaf (a leaf is a node with neither
  a left or right node attached).
• Deletion of a node with one empty subtree (a
  subtree is all nodes below a node with the
  current node as root)
• Deletion of a node where neither subtree is empty.

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Binary Trees - Deleting a node

• Example - Case 1 Deleting a leaf (delete 4)

Becomes
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Binary Trees - Deleting a node

• Example - Case 1 Deleting a leaf (delete 4)

Tree
Container
Root
16
8
24
20
3
27
30
26
1
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Binary Trees - Deleting a node

• Example - Case 2 Deleting empty right subtree
  (delete 8)

Tree
Container
Becomes
Root
16
8
24
20
3
27
30
26
1
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Binary Trees - Deleting a node

• Example - Case 2 Deleting empty right subtree
  (delete 8)

Tree
Container
Root
16
24
3
20
27
1
30
26
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Binary Trees - Deleting a node

• Example - Case 3 Deleting neither subtree is
  empty (delete 24)

Tree
Container
Root
Becomes
16
24
3
20
27
1
30
26
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Binary Trees - Deleting a node

• Example - Case 3 Deleting neither subtree is
  empty (delete 24)

Tree
Container
Root
16
27
3
26
30
1
20
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Summary

• Recursion is a method calling itself.
• Recursion uses the virtual machine stack to keep
  track of program execution.
• Binary trees are a specialized data structures
  dealing with storing data in a specific order.
• Recursion and stacks can be used for traversal
  and manipulation of binary trees.