Go Premium for a chance to win a PS4. Enter to Win

x
?
Solved

recurrence relation for binary tree height algorithm?

Posted on 1998-11-10
9
Medium Priority
?
2,996 Views
Last Modified: 2008-02-26
what would the recurrence relation be for the following pseudo code algorithm?

void Height(T)
{
     if (T is not empty)
     {
          lefttreeht = Height(left subtree of T);
          righttreeht = Height(right subtree of T);
         
          if (lefttreeht > righttreeht)
               return lefttreeht + 1;
          else
               return righttreeht + 1;
     }
     else
          return 0;
}

0
Comment
Question by:melissak
  • 3
  • 2
  • 2
  • +2
9 Comments
 
LVL 3

Expert Comment

by:arnond
ID: 1254160
do you mean big O complexity ?
if so, it's O(log n), n being the number of nodes in the tree.

Arnon David.
0
 

Author Comment

by:melissak
ID: 1254161
No, I mean;

     base case C(0)=0

recurrence relation for a tree of n nodes:

     C(n) = C(n-1) + something or nothing
     C(n+1) = C(n) + something or nothing

*parenthesis indicate subscript

this is what I think the format should somewhat be but
it could be something different.  I do know that the
explicit formula should be:

     C(n)=log(2)[n+1]-1

which was developed from some recurrence relation.
thanks
mk
0
 
LVL 85

Expert Comment

by:ozo
ID: 1254162
Are you assuming the tree is balanced?
0
Concerto's Cloud Advisory Services

Want to avoid the missteps to gaining all the benefits of the cloud? Learn more about the different assessment options from our Cloud Advisory team.

 

Author Comment

by:melissak
ID: 1254163
I don't think it matters.  The definition for the height of a
tree is: the number of nodes on the longest path from the root
to a leaf.  So, worst case binary tree would be one that has nodes attatched to the left or right side only (a diagonal).
0
 
LVL 85

Expert Comment

by:ozo
ID: 1254164
so is C(n) supposed to be the worst case height of a balanced tree of n nodes?
0
 
LVL 16

Accepted Solution

by:
heyhey_ earned 200 total points
ID: 1254165
maybe i am missing something, but if
the explicit formula should be:
     C(n)=log(2)[n+1]-1
than the recurrence relation should be
     C(n+1)=C(n) + log(2)[n+1]
     
0
 

Expert Comment

by:guga
ID: 1254166
I have one comment
if C(n)=log(2)[n+1]-1 then let's n=4

C(n)=log(2)[5]-1;

but height of tree must be integer!
0
 

Expert Comment

by:guga
ID: 1254167
may be (i don't sure)
c(n+1) = C(n) + 1
0
 

Author Comment

by:melissak
ID: 1254168
I had math student help me.  With each call to the algorithm the list of N elements is halved.  Since the algorithm recursively calls itself twice the formula is:
C(N) = 2*C(N/2)
To Prove relation:
Let N = 2^n

C(2^n) = 2*C(2^n-1)
C(2^n-1) = 2*C(2^n-2)
C(2^n) = 2^k * C(2^n-k)
Let k=n
C(2^n) = 2^n * C(2^0) = 2^n

since 2^n = N

C(N) = N

BigOH = O(N)

sorry took so long to respond.  e-mail and internet down.
0

Featured Post

Independent Software Vendors: We Want Your Opinion

We value your feedback.

Take our survey and automatically be enter to win anyone of the following:
Yeti Cooler, Amazon eGift Card, and Movie eGift Card!

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Preface I don't like visual development tools that are supposed to write a program for me. Even if it is Xcode and I can use Interface Builder. Yes, it is a perfect tool and has helped me a lot, mainly, in the beginning, when my programs were small…
Examines three attack vectors, specifically, the different types of malware used in malicious attacks, web application attacks, and finally, network based attacks.  Concludes by examining the means of securing and protecting critical systems and inf…
The goal of this video is to provide viewers with basic examples to understand how to use strings and some functions related to them in the C programming language.
The goal of this video is to provide viewers with basic examples to understand how to create, access, and change arrays in the C programming language.
Suggested Courses

886 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question