1. 1 Data Structures CSCI 132, Spring 2016 Notes 16 Tail Recursion

2. 2 Tail Recursion When the recursive function call is the last thing that occurs in the recursive function, this is called a Tail Recursion. In a tail recursion, there are no pending operations. Any tail recursion can be written as an iteration (a loop).

3. 3 Factorial Revisited int Fact (int n) { if (n <= 0) { return 1; } else { return n * Fact (n - 1); } Is this a tail recursion? No. Can we write this as a tail recursion? Yes! Keep track of the answer in a parameter that gets passed on.

4. 4 Tail Recursion Factorial int factTail (int num, int ans) { if (num <= 0) { return ans; } else { return factTail(num - 1, num*ans); } } int factIter (int n) {//Helper function to initialize ans return factTail(n, 1); }

5. 5 Execution of FactTail numansnum*ansFunction Calls 414Fact(4, 1):24 3412Fact(3, 4):24 21224Fact(2, 12):24 12424Fact(1, 24):24 024 (returned)Fact(0, 24):24

6. 6 Writing factTail as an iteration int factWhile (int n) { }

7. 7 Writing factTail as an iteration int factWhile (int n) { int num = n; int ans = 1; while (num > 0) { ans = ans * num; num = num - 1; } return ans; }

8. 8 Recall Fibonacci

9. 9 Fibonacci in C++ code int Fib( int n ) { if (n < 2) { return n; } else { return Fib(n-1) + Fib(n -2); }

10. 10 Recursive Fibonacci repeats computations

11. 11 Fibonacci as a Tail Recursion nifib(i-1)fib(i)fib(i_next) 5101151011 5211252112 5312353123 5423554235 5535855358 (fib(i) + fib(i-1)) Pass values as parameters of the recursive function: n, i, fibIminus1, fibI (we will work on this in lab).

12. 12 The Tower of Hanoi Setup: A stack of rings in order of largest to smallest is on one of three pegs (largest on the bottom, smallest on the top). Object: Move stack from one peg to another peg following these rules: 1) Cannot put a ring on top of a smaller ring. 2) Can only move 1 ring at a time.

13. 13 Tower of Hanoi Demo A BC

14. 14 Tower of Hanoi Demo A BC

15. 15 Tower of Hanoi Demo A BC

16. 16 Tower of Hanoi Demo A BC

17. 17 Tower of Hanoi Demo A BC

18. 18 Tower of Hanoi Demo A BC

19. 19 Tower of Hanoi Demo A BC

20. 20 Tower of Hanoi Demo A BC

21. 21 Tower of Hanoi Demo A BC

22. 22 Tower of Hanoi Demo A BC

23. 23 Tower of Hanoi Demo A BC

24. 24 Tower of Hanoi Demo A BC

25. 25 Tower of Hanoi Demo A BC

26. 26 Tower of Hanoi Demo A BC

27. 27 Tower of Hanoi Demo A BC

28. 28 Tower of Hanoi Demo A BC

29. 29 Strategy for Tower of Hanoi 1) Move top (n-1) rings from origin to spare (following all the rules). 2) Move bottom ring from origin to destination 3) Move top (n-1) rings from spare to destination (following all the rules).

30. 30 Implementing Tower of Hanoi void hanoi(int n, char start, char end, char spare) { if (n < = 0) { //do nothing } else { hanoi(n-1, start, spare, end); cout << "Move disk " << n << "from " << start << " to " << end << endl; hanoi(n-1, spare, end, start); } } //Outputs sequence of moves to solve tower of hanoi!

31. 31 Invocation Tree for hanoi nodes 2 0 2 12 2 3 2 4 How long will it take to move a tower 100 rings high?