Solved

recurrence relation for binary tree height algorithm?

Posted on 1998-11-10
9
2,931 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
[X]
Welcome to Experts Exchange

Add your voice to the tech community where 5M+ people just like you are talking about what matters.

  • Help others & share knowledge
  • Earn cash & points
  • Learn & ask questions
  • 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 84

Expert Comment

by:ozo
ID: 1254162
Are you assuming the tree is balanced?
0
VIDEO: THE CONCERTO CLOUD FOR HEALTHCARE

Modern healthcare requires a modern cloud. View this brief video to understand how the Concerto Cloud for Healthcare can help your organization.

 

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 84

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 100 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

Industry Leaders: 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

An Outlet in Cocoa is a persistent reference to a GUI control; it connects a property (a variable) to a control.  For example, it is common to create an Outlet for the text field GUI control and change the text that appears in this field via that Ou…
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…
The goal of this video is to provide viewers with basic examples to understand opening and writing to files in the C programming language.
The goal of this video is to provide viewers with basic examples to understand and use switch statements in the C programming language.

615 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