Question

P9.3.1 Determine what the following pseudo-Java method outputs on input n. Prove your answer by induction on the call tree: integer foo (natural n) f if (n <= 1) return n; return foo (n-1) foo (n-2);

Answer 1

Call tree (A recursion tree) is used to avoid duplicate work in recursive calls of the given function. Given pseudo-Java method: Integer foo (natural n) if (n<=1) return n; return foo (n-1) - foo (n-2); From the above pseudo code, foo is a method which accepts the parameter n as a type of natural. The foo method performs an operation in the form of recursive approach. Consider foo (1) = 1, foo (2) = 1 foo (n) = foo (n-1) - foo (n-2). Induction is in the form of a method, which is represented in the below format. Induction hypothesis can be represented in the form of IH(). Let us assume 21:P(n) where n represents the ranges over the all positive integers. P1, P2, Pn and these terms are used in the recursion and induction process is that they can also involve in the recursive calls to foo.

From the above terms P1, P2, Pn, the term Pn represents Pr, for n 21. The complete recursion tree that means call tree of foo (n) is given below. From the above assumptions the recursive tree for n=4 is as follows. foo(4) foo(3) foo(2) foo(2) foo(1) foo(1) foo(0) foo(1) foo(0) call tree of pseudo-Java method.

From the above call tree, the function foo (n) which takes n as input type parameter. • If n = 4 then the function call foo (4) goes to next call as foo (n-1)-foo (n-2) • From the above call foo (4-1) means foo (3) and so on for all calls. The foo () function calls in recursive manner until the foo (n) function returns 1. Finally, the resultant operation performs between the function's foo (n-1) and foo (n-2) like foo (n-1)-foo (n-2) on n number of recursive calls. The base cases of above call tree foo (n) is P (1) and P (2) are true. (P is in the form of induction method). The induction method applied to the resultant call tree, then the process of call goes to one call to another call of the foo (n) function. Induction hypothesis can be represented in the form of IH(j) for j22 is for all n sj(IH represents Induction hypothesis). foo (n) = foo (n-1) - foo (n-2). The resultant term of above statement is foo (n) takes n as input type parameter and performs an operation until the return value 1.