Searching: Binary Trees

1
Searching Binary Trees

• Dr. Yingwu Zhu

2
Review of Linear Search

• Collection of data items to be searched is
  organized in a list x1, x2, xn
• Assume and lt operators defined for the type
• Linear search begins with item 1
• continue through the list until target found
• or reach end of list

3
Linear Search

• Vector based search function
• template lttypename tgt
• void LinearSearch (const vectorlttgt v,
  const t item, boolean found, int
  loc)
• found false loc 0 for ( )
• if (found loc v.size()) return if
  (item vloc) found true else loc

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Linear Search

• Singly-linked list based search function
• template lttypename tgt
• void LinearSearch (NodePointer first, const
  t item, bool found, int loc)
• NodePointer locptrfirst forloc0 !found
  locptr!NULL
• locptrlocptr-gtnext) if (item
  locptr-gtdata) found true
• else loc

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Linear Search
Iterator-based linear search

• template lttypename tgt
• void LinearSearch (const vectorlttgt v , const t
  item, bool found, int loc)
• found false loc 0
• for (vectorlttgtiterator itv.begin() !found
  it ! v.end() it)
• if (it item) found true
• else loc

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Binary Search

• Two requirements?

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Binary Search

• Two requirements
• The data items are in ascending order (can they
  be in decreasing order?? ?)
• Direct access of each data item for efficiency
  (why linked-list is not good!)

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Binary Search

• Binary search function for vector
• template lttypename tgt
• void LinearSearch (const vectorlttgt v,
  const t item, bool found, int
  loc)
• found false loc 0
• int first 0 last v.size() - 1 for (
  )
• if (found first gt last) return loc
  (first last) / 2 if (item lt vloc)
  last loc - 1 else if (item gt vloc)
  first loc 1 else found true // item
  found

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Binary Search vs. Linear Search

• Usually outperforms Linear search O(logn) vs.
  O(n)
• Disadvantages
• Sorted list of data items
• Direct access of storage structure, not good for
  linked-list
• Good news It is possible to use a linked
  structure which can be searched in a binary-like
  manner

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Binary Search Tree

• Consider the following ordered list of integers
• Examine middle element
• Examine left, right sublist (maintain pointers)
• (Recursively) examine left, right sublists

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Binary Search Tree

• Redraw the previous structure so that it has a
  treelike shape a binary tree

12
Trees

• A data structure which consists of
• a finite set of elements called nodes or vertices
• a finite set of directed arcs which connect the
  nodes
• If the tree is nonempty
• one of the nodes (the root) has no incoming arc
• every other node can be reached by following a
  unique sequence of consecutive arcs (or paths)

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Trees

• Tree terminology

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Binary Trees

• Each node has at most two children
• Useful in modeling processes where
• a comparison or experiment has exactly two
  possible outcomes
• the test is performed repeatedly
• Example
• multiple coin tosses
• encoding/decoding messages in dots and dashes
  such as Mores code

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Array Representation of Binary Trees

• Store the ith node in the ith location of the
  array

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Array Representation of Binary Trees

• Works OK for complete trees, not for sparse trees

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Some Tree Definition, p656

• Complete trees
• Each level is completely filled except the bottom
  level
• The leftmost positions are filled at the bottom
  level
• Array storage is perfect for them
• Balanced trees
• Binary trees
• left_subtree right_subtreelt1
• Tree Height/Depth of levels

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Tree Question?

• A complete tree must be a balanced tree?
• Give a node with position i in a complete tree,
  what are the positions of its child nodes?

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Linked Representation of Binary Trees

• Uses space more efficiently
• Provides additional flexibility
• Each node has two links
• one to the left child of the node
• one to the right child of the node
• if no child node exists for a node, the link is
  set to NULL

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Linked Representation of Binary Trees

• Example

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Binary Trees as Recursive Data Structures

• A binary tree is either empty
• or
• Consists of
• a node called the root
• root has pointers to two disjoint binary
  (sub)trees called
• right (sub)tree
• left (sub)tree

Anchor
Which is either empty or
Which is either empty or
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Tree Traversal is Recursive

• If the binary tree is empty thendo nothing
• Else N Visit the root, process dataL Traverse
  the left subtreeR Traverse the right subtree

If the binary tree is empty thendo nothing Else
N Visit the root, process dataL Traverse the
left subtreeR Traverse the right subtree
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Traversal Order

• Three possibilities for inductive step
• Left subtree, Node, Right subtreethe inorder
  traversal
• Node, Left subtree, Right subtreethe preorder
  traversal
• Left subtree, Right subtree, Nodethe postorder
  traversal

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ADT Binary Search Tree (BST)

• Collection of Data Elements
• binary tree
• BST property for each node x,
• value in left child of x lt value in x lt in
  right child of x
• Basic operations
• Construct an empty BST
• Determine if BST is empty
• Search BST for given item

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ADT Binary Search Tree (BST)

• Basic operations (ctd)
• Insert a new item in the BST
• Maintain the BST property
• Delete an item from the BST
• Maintain the BST property
• Traverse the BST
• Visit each node exactly once
• The inorder traversal must visit the values in
  the nodes in ascending order

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BST

• template lttypename Tgt
• class BST public BST() myRoot(0)
  bool empty() return myRootNULL private
  class BinNode public T
  data BinNode left, right
  BinNode() left(0), right(0)
  BinNode(T item) data(item), left(0), right(0)
  //end of BinNode typedef BinNode
  BinNodePtr
• BinNodePtr myRoot

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BST operations

• Inorder, preorder and postorder traverasl
  (recursive)
• Non-recursive traversal

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BST Traversals, p.670

• Why do we need two functions?
• Public void inorder(ostream out)
• Private inorderAux()
• Do the actual job
• To the left subtree and then right subtree
• Can we just use one function?
• Probably NO

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Non-recursive traversals

• Lets try non-recusive inorder
• Any idea to implement non-recursive traversals?
  What ADT can be used as an aid tool?

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Non-recursive traversals

• Basic idea
• Use LIFO data structure stack
• include ltstackgt (provided by STL)
• Chapter 9, google stack in C for more details,
  members and functions
• Useful in your advanced programming

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Non-recursive inorder

• Lets see how it works given a tree
• Step by step

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Non-recursive inorder

• 1. Set current node pointer ptr myRoot
• 2. Push current node pointer ptr and all the
  leftmost nodes on its subtree into a stack until
  meeting a NULL pointer
• 3. Check if the stack is empty or not. If yes,
  goto 5, otherwise goto 4
• 4. Take the top element in stack and assign it to
  ptr, print our ptr-gtdata, pop the top element,
  and set ptr ptr-gtright (visit right subtree),
  goto 2
• 5. Done

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Non-recursive inorder

• template lttypename DataTypegt
• void non_recur_inorder(ostream out)
• if (!myRoot) return // empty BST
• BinNodePointer ptr myRoot
• stackltBSTltDataTypegtBinNodePointergt s //
  empty stack
• for ( )
• while (ptr) s.push(ptr) ptr
  ptr-gtleft // leftmost nodes
• if (s.empty()) break // done
• ptr s.top() s.pop()
• out ltlt ptr-gtdata ltlt
• ptr ptr-gtright // right substree

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Non-recursive traversals

• Can you do it yourself?
• Preorder?
• Postorder?

35
BST Searches

• Search begins at root
• If that is desired item, done
• If item is less, move downleft subtree
• If item searched for is greater, move down right
  subtree
• If item is not found, we will run into an empty
  subtree

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BST Searches

• Write recursive search
• Write Non-recursive search algorithm
• bool search(const T item) const

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Insertion Operation

• Basic idea
• Use search operation to locate the insertion
  position or already existing item
• Use a parent point pointing to the parent of the
  node currently being examined as descending the
  tree

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Insertion Operation

• Non-recursive insertion
• Recursive insertion
• Go through insertion illustration p.677
• Initially parent NULL, locptr root
• Location termination at NULL pointer or
  encountering the item
• Parent-gtleft/right item

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Recursive Insertion

• See p. 681
• Thinking!
• Why using at subtreeroot

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Deletion Operation

• Three possible cases to delete a node, x, from a
  BST
• 1. The node, x, is a leaf

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Deletion Operation

• 2. The node, x has one child

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Deletion Operation

• x has two children

View remove() function
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Question?

• Why do we choose inorder predecessor/successor to
  replace the node to be deleted in case 3?

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Implement Delete Operation

• void delete(const T item) //remove the specified
  item if it exists
• Assume you have this function
• template lttypename Tgt
• void BSTltTgtsearch2(const T item, bool
  found, BSTltTgtBinNodePtr locptr,
  BSTltTgtBinNodePtr parent) const
• locptr myRoot parent NULL
  found false
• while(!found locptr)
• if (item lt locptr-gtdata)
• parent locptr
  locptr locptr-gtleft
• else if (item gt locptr-gtdata)
• parent locptr
  locptr loc-gtright
• else
• found true
• //end while

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Implement Delete Operation

1. Search the specified item on the tree. If not
   found, then return otherwise go to 2.
2. Check if the node to be deleted has two child
   nodes (case 3). If so, find the inorder
   successor/predecessor, replace the data, and then
   make the successor/predecessor node the one to be
   deleted
3. In order to delete the node, you should keep
   track of its parent node in step 1 and 2.
4. Delete the node according to the first two simple
   cases. Be careful with parent
5. Release the memory of the node deleted

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delete(const T item)

• BinNodePtr parent, x
• bool found
• search2(item, found, x, parent) //search
  item on the tree
• if (!found) return
• if (x-gtleft x-gtright) //the case 3 with
  two child nodes
• BinNodePtr xsucc x-gtright
• parent x
• while (xsucc-gtleft) //find inorder
  successor
• parent xsucc xsucc
  xsucc-gtleft
• x-gtdata xsucc-gtdata
• x xsucc
• //end if
• BinNodePtr subtree x-gtleft
• if (!subtree) subtree x-gtright
• if (!parent)
• myRoot subtree
• else
• if (x parent-gtleft)
• parent-gtleft subtree

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BST Class Template

• View complete binary search tree template, Fig.
  12.7
• View test program for BST, Fig. 12.8
• No such functions
• Copy constructor, assignment operator
• So we can NOT do what?

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In-class Exercises 1

• Write a recursive member function height() for
  the BST class template to return the height of
  the BST
• int height()
• // See how concise your implementation could be?

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In-class Exercises 2

• Write a recursive member function leafCount() for
  the BST class template to return the number of
  leaf nodes of the BST
• int leafCount()
• // See how concise your implementation could be?

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Questions?

• What is T(n) of search algorithm in a BST? Why?
• What is T(n) of insert algorithm in a BST?
• Other operations?

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Problem of Lopsidedness

• The order in which items are inserted into a BST
  determines the shape of the BST
• Result in Balanced or Unbalanced trees
• Insert O, E, T, C, U, M, P
• Insert C, O, M, P, U, T, E

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Balanced
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Unbalanced
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Problem of Lopsidedness

• Trees can be totally lopsided
• Suppose each node has a right child only
• Degenerates into a linked list

Processing time affected by "shape" of tree