1. Stacks & Recursion

2. Stack Linear data structure Can only be accessed from one end
Very useful when data has to stored and then retrieved in the reverse order.
Called LAST IN FIRST OUT (LIFO)

3. LIFO : Last In First Out
Stack 10 6 9

4. Stacks
Stack 10 6 9
LIFO : Last In First Out

5. Operations
By Adam Drozdek

6. RECURSIVE STRUCTURE
Sierpinski_Triangle from

7. Recursive Definition
An object is defined in terms of smaller version of itself
A problem is decomposed into simpler subproblems of the same kind
Recursion is the process in which repetition in a “self-similar” manner
Generally used to define functions or sequences of numbers
Has two parts
Base Case / Anchor Case / Ground Case
Define the basic building blocks
Rules / General Case
Define how objects / elements can be built from basic elements or already constructed objects
Serves two purposes
Generating new elements
Testing whether an element belongs to a set

8. Will test if a number is member of series
Examples
Factorials
n! = if n=0
n! = n(n-1)! if n> 0
This definition will evaluate the factorial of a number
The set of natural numbers
0  N
If n  N then (n+1)  N
There are no other natural numbers
This definition will generate natural numbers
Will test if a number is member of series
Will generate numbers

9. Activation Records/Stack Frame
Recursive Function
Recursion means having the characteristic of coming up again and again
A function that invokes itself by making a call to itself is called a recursive function
Direct Recursion: When a function calls itself
Indirect Recursion: When A calls B and B calls A
Activation Records/Stack Frame
The state of each function is characterized by an activation record. It is the private pool of information for a function. It has the following information:
Parameters and local variables: of the function
Dynamic link: Link to the calling function’s activation record
Return Address: Caller function’s return address
Runtime Stack
Runtime stack is a stack whose each element is the activation records of the different functions being called during the running of the program

10. Example int factorial(n) void main() { { int fact = 1;
if (n==0)
return fact;
else
{ fact = n*factorial(n-1);
return fact; }
void main() {
int answer = factorial(3);
cout << answer; }
Factorial(0)
Factorial(1)
Factorial(2)
Factorial(3)
Main()
fact = 1
fact = 1 * factorial(0)
fact = 1*1=1
answer = factorial (3)
fact = 3 * factorial(2)
fact = 2 * factorial(1)
fact = 2 * factorial(1)
answer = factorial (3)
fact = 3 * factorial(2)
fact = 2 * 1=2
fact = 3 * factorial(2)
answer = factorial (3)
fact = 3 * 2 = 6
answer = factorial (3)
answer = 6

11. Towers of Hanoi Watch animation with 4 disks!!!
Pictures and animation from:
void Hanoi(int N,char src,char dest,char intermediate)
{ if (N>0)
{ //move top n-1 disks from ‘src’ to ‘intermediate’
//using ‘dest’as intermediate location
Hanoi(N-1,src,intermediate,dest);
//move the bottom disk from ‘src’ to ‘dest’
Move(src,dest);
//now move the rest of N-1 disks from ‘intermediate’
//to ‘dest’ using ‘src’ as intermediate location
Hanoi(N-1,intermediate,dest,src); }

12. How it is working??? Lets trace for Hanoi(3,’A’,’C’,’B’)
Sequence of moves:
Move(A,B)
Move(AC)
Move(B,C)
Move(B,A)
Move(A,C)
Hanoi(0,A,C,B)
Move(A,B)
Hanoi(0,C,B,A)
Hanoi(1,A,B,C)
Move(A,C)
Hanoi(1,B,C,A)
See animation!!!
Hanoi(2,A,C,B)
Move(A,B)
Hanoi(2,B,C,A)
Hanoi(0,B,A,C)
Move(B,C)
Hanoi(0,A,C,B)
Hanoi(3,A,B,C)
Hanoi(0,B,C,A)
Move(B,A)
Hanoi(0,C,A,B)
Hanoi(1,B,A,C)
Move(B,C)
Hanoi(1,A,C,B)
Hanoi(0,A,B,C)
Move(A,C)
Hanoi(0,B,C,A)

13. Tail Recursion
Use of only one recursive call at the very end of the function implementation. Useful in logic programming languages like prolog.
Non-Tail Recursion
When the recursive call is NOT tail recursion

14. More examples, which you can try…
Fibonacci Series
Linear search
Binary search
Reverse items of an array