Exploring t(n): Recursion Tree Analysis

• scorpius1782
In summary, t(n) = 3t(\frac{n}{2}) + n + 1 for n a power of two greater than 1.The height of the tree is t(n) = 3t(\frac{n}{2}) + n + 1.
scorpius1782

Homework Statement

Let the function t(n) be defined recursively by:

##t(1) = 1##
##t(n) = 3t(\frac{n}{2}) + n + 1## for n a power of two greater than 1.

Draw several levels of the recursion tree for t, and answer the following:

What will the height of the tree be if n is a power of 2?

And what is the sum of values on level 3. assume ##n\geq 16##

The Attempt at a Solution

I did the tree where (n+1) is the root. Each successive level is ##\frac{n+1}{2^k}## with ##3^k## nodes at each height (height =k). So, from my lecture notes to get height I set ##\frac{n+1}{2^k}=1##. I'm sort of stuck here. I do not have the option (it is multiple choice) of ##k=log_2(n+1)## so I must have don't something wrong or I need to manipulate my answer in a way that isn't clear to me.

For the sum of values I have 3^3 nodes each of value ##\frac{n+1}{2^3}## I thought that I could just say ##3^3\frac{n+1}{2^3}## but this is not an answer either. The closest possible answer was ##3^3n+3^3##

Thanks for any help!

Last edited:
scorpius1782 said:

Homework Statement

Let the function t(n) be defined recursively by:

##t(1) = 1##
##t(n) = 3t\frac{n}{2} + n + 1## for n a power of two greater than 1.
I don't understand the line above.

Did you mean this:
##t(n) = 3 t(\frac{n}{2}) + n + 1##
scorpius1782 said:
Draw several levels of the recursion tree for t, and answer the following:

What will the height of the tree be if n is a power of 2?

And what is the sum of values on level 3. assume ##n\geq 16##

The Attempt at a Solution

I did the tree where (n+1) is the root. Each successive level is ##\frac{n+1}{2^k}## with ##3^k## nodes at each height (height =k). So, from my lecture notes to get height I set ##\frac{n+1}{2^k}=1##. I'm sort of stuck here. I do not have the option (it is multiple choice) of ##k=log_2(n+1)## so I must have don't something wrong or I need to manipulate my answer in a way that isn't clear to me.

For the sum of values I have 3^3 nodes each of value ##\frac{n+1}{2^3}## I thought that I could just say ##3^3\frac{n+1}{2^3}## but this is not an answer either. The closest possible answer was ##3^3n+3^3##

Thanks for any help!

Yes sorry, i fixed it. I pasted plain text and took them out for some reason.

I would start playing with it. You're given t(1) = 1. What's t(2)? t(4)? And so on. I'm not sure what is meant by "recursion tree" other than how many times the formula will be used to evaluate t(n).

I would first like to clarify that recursion tree analysis is a method used to analyze the time complexity of recursive functions, where the height of the recursion tree represents the number of recursive calls made and the number of nodes at each level represents the number of operations performed at that level.

Now, let's look at the function t(n) given in the homework statement. If n is a power of 2, then we can write n as ##2^k## for some positive integer k. In this case, the recursion tree will have k levels, since each time the function is called, the input is halved. So, the height of the tree will be k, which is equal to log2(n).

To find the sum of values on level 3, we need to first determine the number of nodes at that level. Since the input is halved at each recursive call, the number of nodes at level 3 will be ##\frac{n}{2^3} = \frac{n}{8}##. Now, each node at this level will have a value of ##n+1##, so the sum of values on level 3 will be ##\frac{n}{8}(n+1) = \frac{n^2+n}{8}##. Therefore, the correct answer would be ##\frac{n^2+n}{8}##.

1. What is recursion tree analysis?

Recursion tree analysis is a technique used in computer science and mathematics to analyze recursive algorithms. It involves drawing a tree diagram to represent the recursive calls made by an algorithm and then analyzing the number of nodes and levels in the tree to determine the time complexity of the algorithm.

2. How is a recursion tree constructed?

To construct a recursion tree, start with the initial call of the recursive function at the top of the tree. Then, for each recursive call, draw a new branch on the tree. Repeat this process until all recursive calls have been made and the tree is complete.

3. What does the number of levels in a recursion tree represent?

The number of levels in a recursion tree represents the number of times the recursive function is called. This can give insight into the time complexity of the algorithm, as each level typically represents a single iteration or recursive call.

4. How can recursion tree analysis help in algorithm design?

Recursion tree analysis can help in algorithm design by providing a visual representation of the recursive calls and allowing for the identification of potential inefficiencies or optimizations. It can also help in determining the time complexity of an algorithm and comparing it to other approaches.

5. What are the limitations of recursion tree analysis?

Recursion tree analysis may not be suitable for all types of recursive algorithms, as some may have variable or unpredictable recursive calls. It also does not take into account other factors such as memory usage or any optimizations made within the algorithm. Additionally, it may not accurately represent the actual time complexity of an algorithm in all cases.

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