Searching via Traversals Searching a Binary Search Tree (BST) Binary Search on a Sorted Array Data Structure Conversion and Helper Modules

Searching via Traversals Searching a Binary Search Tree (BST)

Binary Search on a Sorted Array Data Structure Conversion

and Helper Modules

Plan of Attack

• Simple searching just involves traversing a data structure until the data element is found.

• The only twist is that if the collection is ordered (like a linked lists) you can stop if you don’t find an element and you pass the point where it should be located.

• So we’re going to skip numerous slides which essentially review material we’ve already covered.

• So we’re going to focus on Binary Search and Data Structure Conversion

• But first the twist...

LB

procedure Search ( cur isoftype in ptr toa Node, target isoftype in Num )

loop exitif((cur = NIL) OR (cur^.data = target)) cur <- cur^.next endloop

if(cur = NIL) then print(“Target data not found”) else print(“Target data found”) endifendprocedure // Search

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okToExit isoftype Boolean okToExit <- FALSE loop exitif(okToExit) if(cur = NIL OR cur^.data > target) then okToExit <- TRUE print(“Target data not found”) elseif(cur^.data = target) then okToExit = TRUE print(“Found target”) // Could transfer data to param here else

cur <- cur^.next endif endloop

endprocedure // Search

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Same logic can be applied to Trees

Skipping stuff starting on Page 120!

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Searching via Traversals

An Example

• Imagine we have a database of students• I want to know if Bob Smith is in my

class.

• I want to search my database and have the module print out “IN THE CLASS” or “NOT IN THE CLASS.”

Another Situation

• I have the same student database.• I need to view Alice Jones’ grades

and update them.

• Again, I need to search the database.• This time, I need Alice’s information

returned to me so I can view and modify them (i.e. need access).

The Scenario

• We have some collection(stored in an array, tree, or list)–May be sorted or not–May be empty or not

• We want the determine if a particular element is in the collection or not.

• We need to search for the element.

Searching

• Given the collection and an element to find…

• Determine whether the “target” element was found in the collection– Print a message– Return a value

(an index or pointer, etc.)• Don’t modify the collection in the

search!

Key Fields

• Often, our collections will hold complex structures (i.e. have many items of information in each).

• It is common to organize our collection using one “part” (or field)– Name– Student Number

• This is called the “key field”• Then we can search on the key field.

A Simple Search

• A search traverses the collection until– The desired element is found – Or the collection is exhausted

• If the collection is ordered, I might not have to look at all elements– I can stop looking when I know

the element cannot be in the collection.

Searching in an Unordered Collection

• Let’s determine if the value 12 is in the collection:

35 42 12 5 \\

12 Found!

Head

Searching in an Unordered Collection


35 42 12 5 \\

13 Not Found!

Head

Searching in an Ordered Collection


5 12 35 42 \\

13 Not Found!

Head

Search Examples

Arrays

Linked Lists

Binary Trees

Search Example on a List

An Iterative Search

Let’s build a module to search a list of numbers using iteration.

We’ll print whether the number was found or not.

Steps:– Traverse until we find a match

or reach the end.– Print the results.

An Iterative Un-Ordered Searchprocedure Search ( ??? ) loop ??? exitif( ??? ) ??? endloop

if( ??? ) then print(“Target data not found”) else print(“Target data found”) endifendprocedure // Search

An Iterative Un-Ordered Searchprocedure Search ( cur isoftype in ptr toa Node,

target isoftype in Num ) loop ??? exitif( ??? ) ??? endloop



target isoftype in Num ) loop exitif( cur = NIL ) cur <- cur^.next endloop



target isoftype in Num ) loop exitif((cur = NIL) OR (cur^.data = target)) cur <- cur^.next endloop




if( cur = NIL ) then print(“Target data not found”) else print(“Target data found”) endifendprocedure // Search




What’s Different for An Iterative Ordered Search?




An Iterative Ordered Searchprocedure Search ( cur isoftype in ptr toa Node,

target isoftype in Num ) loop exitif((cur = NIL) OR (cur^.data >= target)) cur <- cur^.next endloop

if((cur = NIL) OR (cur^.data > target)) then print(“Target data not found”) else print(“Target data found”) endifendprocedure // Search

Tracing the Search

Now let’s call the un-ordered version of the module:

algorithm Example

. . .

Search(head, 4)

. . .

endalgorithm // Example

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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur




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target = 4cur

Target data found




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target = 4cur

Tracing the Ordered Search

Now let’s call the ordered version of the module:

algorithm Example

. . .

Search(head, 9)

. . .

endalgorithm // Example

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loop exitif((cur = NIL) OR (cur^.data >= target)) cur <- cur^.next endloop


target = 9cur

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target = 9cur

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target = 9cur

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target = 9cur

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target = 9cur

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target = 9cur

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target = 9cur

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Target data not found




target = 9cur

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Searching an Array

• Same principles apply:– Examine cells until we find a match

or exhaust the possibilities– Then indicate whether the item was

in the collection (print or return information)

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Un-Ordered Iterative Array Search

procedure Search(my_array isoftype in NumArrayType, target isoftype in Num) i isoftype Num i <- 1 loop exitif((i > MAX) OR (my_array[i] = target)) i <- i + 1 endloop

if(i > MAX) then print(“Target data not found”) else print(“Target data found”) endifendprocedure // Search

Calling the Module

algorithm Example2

MAX is 6

. . .

Search(MyNumArray, 13)

. . .

endalgorithm // Example2



my_array7 12 5 22 13 32

1 2 3 4 5 6target = 13



my_array7 12 5 22 13 32

1 2 3 4 5 6target = 13



my_array7 12 5 22 13 32

1 2 3 4 5 6target = 13



my_array7 12 5 22 13 32

1 2 3 4 5 6target = 13



my_array7 12 5 22 13 32

1 2 3 4 5 6target = 13



my_array7 12 5 22 13 32

1 2 3 4 5 6target = 13



my_array7 12 5 22 13 32

1 2 3 4 5 6target = 13



my_array7 12 5 22 13 32

1 2 3 4 5 6target = 13

Target data found



my_array7 12 5 22 13 32

1 2 3 4 5 6target = 13

Binary Tree Searches

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Searching With Binary Trees

Have to visit every node in the binary tree.

3 types of traversals:

Preorder Inorder Postorder

Check Node

Go left

Go right

Go left

Check Node

Go right

Go left

Go right

Check Node

Pre-Order Search Traversal Algorithm

• As soon as we get to a node, check to see if we have a match

• Otherwise, look for the element in the left sub-tree

• Otherwise, look for the element in the right sub-tree

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Left ??? Right ???

Search on a Binary Tree

function BTSearch returnsa Boolean

(cur iot in ptr toa Node, toFind iot in Num)

if (cur = NIL) then

BTSearch returns false

elseif (cur^.data = toFind) then

BTSearch returns true

else

BTSearch returns BTSearch(cur^.left, toFind)

OR BTSearch(cur^.right, toFind)

endif

endfunction // BTSearch

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Find 14

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Find 14

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Find 14

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Find 14

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Find 14

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Find 14...Found!!!

cur

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Search on a Binary Tree

function BTSearch returnsa Boolean

(cur iot in ptr toa Node, toFind iot in Num)

if (cur = NIL) then

BTSearch returns false

elseif (cur^.data = toFind) then

BTSearch returns true

else

BTSearch returns BTSearch(cur^.left, toFind)

OR BTSearch(cur^.right, toFind)

endif

endfunction // BTSearch

LB

Once we’ve found the 14 dowe search the right subtree?

Summary

• Searches involve visiting elements in a collection until– A match is found– Or until all possible locations are exhausted

• With sorted collections, we may be able to “stop early”

• Once we determine if the element is in the collection, then we do some work:– Print the results– Return some information (pointer, index)

Questions?

Searching a Binary Search Tree (BST)

The Scenario

• We’ve got a Binary Search Tree and we want to determine if an element is in the collection.

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LessThan

14

GreaterThan

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Cutting the Work in Half

• In searching for a match, we can ignore half of the tree at each comparison.

14

LessThan

14

Looking for 42…

It must be to theright!

GreaterThan

14

The Binary Search Algorithm

if at NIL // not found

DO NOT FOUND WORK

elseif ( match ) then // found

DO FOUND WORK

elseif ( value to match < current value )

recurse left // must be to left

else

recurse right // must be to right

The Binary Search for a BST

procedure Search(cur iot in Ptr toa Node, target isoftype in Num) if(cur = NIL) then print(“Not Found”) elseif(cur^.data = target) print(“Target Found”) elseif(cur^.data > target) Search(cur^.left, target) else Search(cur^.right, target) endifendprocedure // Search

head

42

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.Search(head, 35)Search(head, 87)..

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head

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head

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Procedure Search( cur iot in Ptr toa Node, target iot in num) if(cur = NIL) then print(“Not Found”) elseif(cur^.data = target) print(“Target Found”) elseif(cur^.data > target) Search(cur^.left, target) else Search(cur^.right, target) endifendprocedure // Search

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target = 35

cur

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head

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47

target = 35

cur

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head

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target = 35

cur

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head

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target = 35

cur

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head

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target = 35


cur

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head

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target = 35


cur

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target = 35


cur

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target = 35


cur

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target = 35



cur

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head

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target = 35



cur

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head

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Target Found

cur

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head

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cur

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head

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cur

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head

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cur

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target = 87

cur

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head

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target = 87

cur

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head

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target = 87

cur

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head

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target = 87

cur

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target = 87


cur

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head

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target = 87


cur

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head

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target = 87


cur

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head

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target = 87


cur

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head

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target = 87



cur

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head

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Not Found

cur

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head

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cur

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cur

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cur

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Summary

• We can cut the work in half after each comparison.

• Recurse in one direction – left or right.

• When we reach NIL, then we no the value wasn’t in the binary search tree.

Questions?

Binary Search on a Sorted Array

The Scenario

• We have a sorted array• We want to determine if a particular

element is in the array– Once found, print or return

(index, boolean, etc.)– If not found, indicate the element

is not in the collection

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A Better Search

• Of course we could use our simpler search and traverse the array

• But we can use the fact that the array is sorted to our advantage

• This will allow us to reduce the number of comparisons

Binary Search

• Requires a sorted array or a binary search tree.

• Cuts the “search space” in half each time.• Keeps cutting the search space in half

until the target is found or has exhausted the all possible locations.

• Inappropriate for linked lists because linked lists lack random access.

The Algorithm

look at “middle” element

if no match then

look left or right

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The Algorithm


if no match then

look left or right

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The Algorithm


if no match then

look left or right

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The Algorithm


if no match then

look left or right

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• Return found or not found (true or false), so it should be a function.

• We’ll use recursion• We must pass the entire array into the

module each pass, so set bounds (search space) via index bounds (parameters)–We’ll need a first and last


calculate middle position

if (first and last have “crossed”) then

DO NOT FOUND WORK

elseif (element at middle = to_find) then

DO FOUND WORK

elseif to_find < element at middle then

Look to the left

else

Look to the right

Looking Left

• Use indices “first” and “last” to keep track of where we are looking

• Move left by setting last = middle – 1

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F LML

Looking Right

• Use indices “first” and “last” to keep track of where we are looking

• Move right by setting first = middle + 1

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F LM F

Binary Search Example – Found

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Looking for 42

F LM


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Looking for 42

F LM


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42 found – in 3 comparisons

F

LM

Binary Search Example – Not Found

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Looking for 89

F LM


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Looking for 89

F LM


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Looking for 89

F L

M


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89 not found – 3 comparisons

FL

Function Find returnsa boolean (A isoftype in ArrayType, first, last, to_find isoftype in num)

// contract (Pre, Post, Purpose) here

middle isoftype num

middle <- (first + last) div 2

if (first > last) then

Find returns false

elseif (A[middle] = to_find) then

Find returns true

elseif (to_find < A[middle]) then

Find returns Find(A, first, middle–1, to_find)

else

Find returns Find(A, middle+1, last, to_find)

endfunction

Summary

• Binary search reduces the work by half at each comparison

• With the array, we need to keep track of which “part” is currently “visible”

• We know the element isn’t in the array when our first and last indices “cross”

Questions?

Data Structure Conversionand Helper Modules

The Scenario

• Imagine you have a collection of data stored in a linked list.

• But now you need to store this data in a binary search tree.

• You’ll need to convert the data from one structure to the other without altering the data itself.

An Example: List to Tree

Purpose of Structure Conversion

Given the input data arranged in one format:

• Keep the original data andstructure intact

• Return the same data, but arrangedin a different order or within a different data structure

Conversion Examples

• Linked List to a Binary Search Tree• Binary Search Tree to a Sorted Linked List• Array to a Linked List• Array to a Binary Search Tree

Any others are also valid, but going from a dynamic structure to a static structure may run out of allocated space.

Steps Involved

• Initialize the new structure• Traverse the original structure– At each element, insert the current

element into the new structure• Return the new structure

Note that the original structure and data is unchanged.

An Example: List to Tree

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Initialize the New Structure

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Traverse the Original Structure and Insert into

the New Structure

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the New Structure

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Done TraversingNow Return the New Structure

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Needed Modules

• A module to initialize the new structure• A module to traverse the initial structure • A module to insert an element into the

new structure

• But all of this should betransparent to the user!

Helper Modules

• In order to correctly perform some tasks, helper modules are often necessary.

• Helper (or wrapper) modules are other modules that help in the completion of a task.

• They are often useful in hiding details.

The Conversion Helper Modules

Convert

OriginalStructure

NewStructure

The Conversion Helper Modules

Convert

Initialize

OriginalStructure

NewStructure

Traverse Original

Insert into New

A First Try at the Conversion Module

procedure Tree_from_List(cur isoftype in Ptr toa List_Node, head isoftype in/out Ptr toa Tree_Node )

// Pre: head initialized to NIL

if( cur <> NIL ) then

BST_Insert(head, cur^.data )

Tree_from_List(cur^.next, head )

endif

endprocedure

A First Try at the Algorithm

algorithm ListToTree // some delcarations TreePtr isoftype Ptr toa Tree_Node ListPtr isoftype Ptr toa List_Node // build the list work here TreePtr <- NIL Tree_from_List( ListPtr, TreePtr ) // now the tree is builtendalgorithm // ListToTree

The user is required to initialize the list pointer.It is better to wrap the recursive call into a user

interface program as follows…

A Better Main Algorithm

The initialization should be hidden inside the conversion module itself.

algorithm BetterListToTree TreePtr isoftype Ptr toa Tree_Node ListPtr isoftype Ptr toa List_Node

// build the list work is here

Nice_Tree_from_List( ListPtr, TreePtr ) // now the tree is built

endalgorithm // BetterListToTree

The “Wrapper” Module

procedure Nice_Tree_from_List( cur isoftype in Ptr toa List_Node, head isoftype out Ptr toa Tree_Node )

// perform the initialization head <- NIL

// build the list Tree_From_List( cur, head )endprocedure

Summary

• Keep the original structure unchanged• Initialize the new structure• Traverse the old structure– Insert each element into the new structure

• Helper (wrapper) modules– Help hide details– Properly abstract tasks into

separate modules

Questions?

Documents

Searching via Traversals Searching a Binary Search Tree (BST) Binary Search on a Sorted Array Data Structure Conversion and Helper Modules