1. Recursion
CSCE 121
J. Michael Moore

2. Self-Similarity
Romanesco broccoli

3. Self-Similarity

4. Fractals
Repeated self-similarity can be infinite.
Koch Snowflake

5. Landscapes
Add randomness to repeated pattern
landscapes.

6. Fractal Landscape
By The Ostrich - Own work, CC BY 3.0,

7. Mathematically
4!= 4× 4−1 != 4 ×(3!)= 4×(3× 3−1 !)= 4×(3× 2! )= 4×(3× 2× 2−1 ! )= 4×(3× 2× 1! )= 4×(3× 2× 1× 1−1 ! )= 4×(3× 2× 1× 0! )= 4×(3× 2× 1× 1 )= 24
A function defined in terms of itself.
Factorial: 𝑛!=𝑛× 𝑛−1 !
Where:
0!=1
Recursive Call.
Base case.

8. Recurrence Relationships
Fibonacci Numbers:
𝐹 𝑛 = 𝐹 𝑛−1 + 𝐹 𝑛−2
Where:
𝐹 0 =0 and 𝐹 1 =1
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …
Recursive Calls.
Base cases.
𝐹 0
𝐹 1
𝐹 2
𝐹 3
𝐹 4
𝐹 5
𝐹 6
𝐹 7
𝐹 8
𝐹 9
𝐹 10
𝐹 11
𝐹 12
𝐹 13
𝐹 14 1 2 3 5 8 13 21 34 55 ?

9. Computer Science A function that calls itself.
Strongly related to mathematical definition.
Think about induction if you have taken a discrete math class.
Examples later

10. Factorial Flowchart
Self Recursion.
Base case.
Recursive Call.

11. Indirect/Mutual Recursion.
Contrived Example
Indirect/Mutual Recursion.

12. Challenges
What if you do not include a base case or get to the base case?
Infinite recursion (think of an infinite loop)
Each time the recursive call occurs, a new stack frame is created…
Eventually, you will run out of memory!
Stack Overflow

13. Rules for Good Recursive Function
One or more base cases
Always returns without further recursion
One or more recursive function calls
Must get closer to base case
Don‘t forget to use the result of your recursive call!

14. Iteration Recursion is a type of iteration!
Frequently, we use recursion because it looks more like the real problem (e.g. Fibonacci numbers).
Every time you call a function there is overhead. Creating stack frame, saving address to return to, etc.

15. Recursion vs. Looping
Generally, anything that is written recursively can be converted into a loop. Sometimes, the looping version is more efficient, sometimes not.
You will see more useful applications of recursion in 221.
Tree and graph traversals, etc.
However, we want you to have some exposure now, so you do not have to learn recursion and tackle a harder problem at the same time.