Recursive algorithms in computer science courses: Fibonacci numbers and binomial coefficients

We observe that the computational inefficiency of branched recursive functions was not appropriately covered in almost all textbooks for computer science courses in the first three years of the curriculum. Fibonacci numbers and binomial coefficients were frequently used as examples of branched recursive functions. However, their exponential time complexity was rarely claimed and never completely proved in the textbooks. Alternative linear time iterative solutions were rarely mentioned. We give very simple proofs that these recursive functions have exponential time complexity. The proofs are appropriate for coverage in the first computer science course.