Recursion Recursion is a math and programming tool –Technically, not necessary Advantages of recursion –Some things are very easy to do with it, but difficult to do without it –Frequently results in very short programs/algorithms Disadvantages of recursion –Somewhat difficult to understand at first –Often times less efficient than non-recursive counterparts –Presents new opportunities for errors and misunderstanding –Tempting to use, even when not necessary Recommendation – use with caution, and only if helpful => Read section 2.3

Recursive Mathematical Definitions Factorial - Non-Recursive Definition N! = N * (N-1) * (N-2) * … * 2 * 1 *Note that a corresponding Java program is easy to write public static int fact(int n) :

Factorial - Recursive Definition 1if N=1Basis Case N * (N-1)!if N>=2Recursive Case Why is it called recursive? Why do we need a basis case? Note that the “recursive reference” is always on a smaller value. N! = {

Fibonacci - Non-Recursive Definition … *Note that a corresponding Java program is easy to write…or is it? public static int fib(int n) :

Fibonacci - Recursive Definition 0if N=1Basis Case 1if N=2Basis Case fib(N-1) + fib(N-2)if N>=3Recursive Case Note there are two basis cases and two recursive references. fib(N) = {

Recursive Java Programs Printing N Blank Lines – Non-Recursive public static void NBlankLines(int n) { for (int i=1; i<=n; i++) System.out.println(); }

Printing N Blank Lines – Recursive // NBlankLines outputs n blank lines, for n>=0 public static void NBlankLines(int n) { if (n <= 0)Basis Case return; else { System.out.println(); NBlankLines(n-1);Recursive Case } *Don’t ever write it this way; this is a simple, first example of recursion.

Another Equivalent Version (slightly restructured) // NBlankLines outputs n blank lines, for n>=0 public static void NBlankLines(int n) { if (n > 0) { System.out.println(); NBlankLines(n-1); }

public static void main(String[] args) { : NBlankLines(3); : } public static void NBlankLines(int n) {n=3 if (n > 0) { System.out.println(); NBlankLines(n-1); } } public static void NBlankLines(int n) {n=2 if (n > 0) { System.out.println(); NBlankLines(n-1); } } public static void NBlankLines(int n) {n=1 if (n > 0) { System.out.println(); NBlankLines(n-1); } } public static void NBlankLines(int n) {n=0 if (n > 0) { System.out.println(); NBlankLines(n-1); } }

A Similar Method: public static void TwoNBlankLines(int n) { if (n > 0) { System.out.println(); TwoNBlankLines(n-1); System.out.println(); }

public static void main(String[] args) { : TwoNBlankLines(2); : } public static void TwoNBlankLines(int n) {n=2 if (n > 0) { System.out.println(); TwoNBlankLines(n-1); System.out.println(); } } public static void TwoNBlankLines(int n) {n=1 if (n > 0) { System.out.println(); TwoNBlankLines(n-1); System.out.println(); } } public static void TwoNBlankLines(int n) {n=0 if (n > 0) { System.out.println(); TwoNBlankLines(n-1); System.out.println(); } }

Are the Following Methods the Same or Different? public static void TwoNBlankLines(int n) { if (n > 0) { System.out.println(); TwoNBlankLines(n-1); } public static void TwoNBlankLines(int n) { if (n > 0) { TwoNBlankLines(n-1); System.out.println(); }

Recursive Factorial Definition 1if N=1Basis Case N * (N-1)!if N>=2Recursive Case Recursive Factorial Program public static int fact (int n) { if (n==1) return 1;Basis Case else { int x;Recursive Case x = fact (n-1); return x*n; } } N! = {

Another Version public static int fact (int n) { if (n==1) return 1;Basis Case else return n*fact (n-1);Recursive Case }

Recursive Fibonacci Definition 0if N=1Basis Case 1if N=2Basis Case fib(N-1) + fib(N-2)if N>=3Recursive Case Recursive Fibonacci Program public static int fib (int n) { if (n==1) return 0;Basis Case else if (n==2) return 1; Basis Case else { int x,y;Recursive Case x = fib (n-1); y = fib (n-2); return x+y; } } fib(N) = {

Another Version public static int fib (int n) { if (n==1) return 0; else if (n==2) return 1; else return fib(n-1) + fib(n-2); }

Recursion & The Run-Time Stack How does recursion related to stack frames and the run time stack? –Note that stack frames are sometimes called allocation records or activation records Why might a recursive program be less efficient than non-recursive counterpart? Why is the recursive fibonnaci function especially inefficient?