How many recursive calls does a recursive function make?

Abstract

The calculation of the Fibonacci sequence using recursion gives rise to an interesting question: How many times does a recursive function call itself? This paper presents one way to examine this question using difference equations with initial conditions, or discrete dynamical systems (DDS). We show that there is a linear relationship between the Fibonacci numbers themselves and the number of recursive calls. This relationship generalizes to any type of DDS of second-order, and DDS of higher-order.

On the number of recursive calls of recursive functions

“…This way, we have reached the same result with [2], however, in a much simpler way from the mathematical and pedagogical point of view.…”

“…Motivation to this report was the article by Robertson [2] on the number of recursive calls of a recursive formula. Here we re-examine the problem using a different and much simpler approach.…”

“…The author in [2] does not solve this recurrence but he connects this recurrence with the following recurrence:…”

“…In many occasions, recursive formulae lead to recursive functions/procedures that are highly inefficient as calls with the same parameters are executed several times. Here, we elaborate on a previous report [2], where a generalized analysis is carried out to derive the number of recursive calls of a recursive formula, the calculation of the Fibonacci numbers in particular. Here we re-examine the problem using a different and simpler approach, which generalizes as well.…”

“…Motivation of the present report is the article of John Robertson in the June 1999 issue of Inroads [2], where an analysis is carried out to derive the number of recursive calls of a recursive formula using difference equations with initial conditions, or discrete dynamical systems (DDS). The author shown that there is a linear relationship between the Fibonacci numbers themselves and the number of recursive calls.…”