1. Data Structures Using C++1 Chapter 6 Recursion

2. Data Structures Using C++2 Chapter Objectives Learn about recursive definitions Explore the base case and the general case of a recursive definition Discover recursive algorithms Learn about recursive functions Explore how to use recursive functions to implement recursive algorithms Learn about recursion and backtracking

3. Data Structures Using C++3 Recursive Definitions Recursion –Process of solving a problem by reducing it to smaller versions of itself Recursive definition –Definition in which a problem is expressed in terms of a smaller version of itself –Has one or more base cases

4. Data Structures Using C++4 Recursive Definitions Recursive algorithm –Algorithm that finds the solution to a given problem by reducing the problem to smaller versions of itself –Has one or more base cases –Implemented using recursive functions Recursive function –function that calls itself Base case –Case in recursive definition in which the solution is obtained directly –Stops the recursion

5. Data Structures Using C++5 Recursive Definitions General solution –Breaks problem into smaller versions of itself General case –Case in recursive definition in which a smaller version of itself is called –Must eventually be reduced to a base case

6. Data Structures Using C++6 Tracing a Recursive Function Recursive function –Has unlimited copies of itself –Every recursive call has its own code own set of parameters own set of local variables

7. Data Structures Using C++7 Tracing a Recursive Function After completing recursive call: Control goes back to calling environment Recursive call must execute completely before control goes back to previous call Execution in previous call begins from point immediately following recursive call

8. Data Structures Using C++8 Recursive Definitions Directly recursive: a function that calls itself Indirectly recursive: a function that calls another function and eventually results in the original function call Tail recursive function: recursive function in which the last statement executed is the recursive call Infinite recursion: the case where every recursive call results in another recursive call

9. Data Structures Using C++9 Designing Recursive Functions Understand problem requirements Determine limiting conditions Identify base cases

10. Data Structures Using C++10 Designing Recursive Functions Provide direct solution to each base case Identify general case(s) Provide solutions to general cases in terms of smaller versions of itself

11. Data Structures Using C++11 Recursive Factorial Function int fact(int num) { if(num == 0) return 1; else return num * fact(num – 1); }

12. Data Structures Using C++12 Recursive Factorial Trace

13. Data Structures Using C++13 Recursive Implementation: Largest Value in Array int largest(const int list[], int lowerIndex, int upperIndex) { int max; if(lowerIndex == upperIndex) //the size of the sublist is 1 return list[lowerIndex]; else { max = largest(list, lowerIndex + 1, upperIndex); if(list[lowerIndex] >= max) return list[lowerIndex]; else return max; }

14. Data Structures Using C++14 Execution of largest(list, 0, 3)

15. Data Structures Using C++15 Recursive Fibonacci int rFibNum(int a, int b, int n) { if(n == 1) return a; else if(n == 2) return b; else return rFibNum(a, b, n - 1) + rFibNum(a, b, n - 2); }

16. Data Structures Using C++16 Execution of rFibonacci(2,3,5)

17. Data Structures Using C++17 Towers of Hanoi Problem with Three Disks

18. Data Structures Using C++18 Towers of Hanoi: Three Disk Solution

19. Data Structures Using C++19 Towers of Hanoi: Three Disk Solution

20. Data Structures Using C++20 Towers of Hanoi: Recursive Algorithm void moveDisks(int count, int needle1, int needle3, int needle2) { if(count > 0) { moveDisks(count - 1, needle1, needle2, needle3); cout<<"Move disk "<<count<<“ from "<<needle1 <<“ to "<<needle3<<"."<<endl; moveDisks(count - 1, needle2, needle3, needle1); }

21. Data Structures Using C++21 Decimal to Binary: Recursive Algorithm void decToBin(int num, int base) { if(num > 0) { decToBin(num/base, base); cout<<num % base; }

22. Data Structures Using C++22 Execution of decToBin(13,2)

23. Data Structures Using C++23 Recursion or Iteration? Two ways to solve particular problem –Iteration –Recursion Iterative control structures: uses looping to repeat a set of statements Tradeoffs between two options –Sometimes recursive solution is easier –Recursive solution is often slower

24. Data Structures Using C++24 Backtracking Algorithm Attempts to find solutions to a problem by constructing partial solutions Makes sure that any partial solution does not violate the problem requirements Tries to extend partial solution towards completion

25. Data Structures Using C++25 Backtracking Algorithm If it is determined that partial solution would not lead to solution –partial solution would end in dead end –algorithm backs up by removing the most recently added part and then tries other possibilities

26. Data Structures Using C++26 Chapter Summary Recursive definitions Recursive algorithms Recursive functions Base cases General cases

27. Data Structures Using C++27 Chapter Summary Tracing recursive functions Designing recursive functions Varieties of recursive functions Recursion vs. Iteration Backtracking