More Recursion

Intro to Computer Science

CS1510

Dr. Sarah Diesburg

Recursion – one way

A class of methods exhibit recursive behavior when they can be defined by two properties: A simple base case (or cases), and A set of rules which reduce all other cases toward

the base case. (recursive step)

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Recursion – another way

A recursive function is: One that calls itself

With “smaller” inputs (a recursive step) Until those inputs reach a base case

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A language based example… The following is a recursive definition of a

person's ancestors: Your parents are your ancestors (base case). The parents of your ancestors are also your

ancestors (recursion step).

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Recursion Humor

Recursion See "Recursion".

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Better Recursion Humor

Recursion If you still don't get it, see "Recursion".

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Recursion Humor

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Some Examples of things that can be done recursively Summation of numbers – sum(lower,upper) Exponents - power(base,exp) Reverse a string – reverse(string) Merge Sort – mergeSort(lyst)

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How Does the Computer Keep Track?

The Stack

A Stack is a data structure, like a List or a Dictionary, but with a few different characteristics.

A Stack is a sequence. A Stack only allows access to one end of its

data, the top of the stack.

Operations

pop: remove top of stack. Stack is one element smaller.

push (val): add val to the stack. Val is now the top. Stack is one element larger.

top: Reveals the top of the stack. No modification to stack.

Stack of Function Calls

Python maintains a stack of function calls. If a function calls another function recursively,

the new function is pushed onto the calling stack and the previous function waits.

The top is always the active function. When a pop occurs, the function below

becomes active.

Example: Fibonacci Sequence Start with two numbers in the sequence of

1 1 To get the next number in the sequence, add

the previous two numbers together 1+1=2

1 1 2 1+2=3

1 1 2 3

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How to make Fibonacci a Recursive Function Depends on the Fibo results for the two

previous values in the sequence. The base values are Fibo(0) == 1 and Fibo(1)

== 1.

fibo (x) = fibo(x-1) + fibo(x-2)

Code Listing 16-3

def fibo(n):

"""Recursive Fibonacci sequence."""

if n == 0 or n == 1: # base case

return 1

else:

# divide and conquer

return fibonacci(n-1) + fibonacci(n-2)

Trace

fibo(4) = fibo(3) + fibo(2) fibo(3) = fibo(2) + fibo(1) fibo(2) = fibo(1) + fibo(0) = 2 # base case fibo(3) = 2 + fibo(1) = 3 # base case fibo(4) = 3 + 2 = 5