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What role does the base case f(1)=f(2)=1f(1) = f(2) = 1 play in this analysis?

The base case f(1)=f(2)=1f(1) = f(2) = 1 determines where the recursion stops, meaning certain branches halt splitting at the bottom level. While this allows the instructor to omit drawing the entire tree in practice, the theoretical complexity analysis still estimates the scale based on the full binary tree's node count.

Conditions

  • The recurrence relation is f(n)=f(n−1)+f(n−2)f(n) = f(n-1) + f(n-2).
  • The base cases are f(1)=1f(1) = 1 and f(2)=1f(2) = 1.
  • Complexity analysis uses a full binary tree for theoretical estimation.

Reasoning, step by step

  1. Identify the base cases f(1)=1f(1) = 1 and f(2)=1f(2) = 1.
  2. Recognize that branches stop splitting when they reach these base cases.
  3. Note that the full tree does not need to be drawn in practice due to this truncation.
  4. Understand that the theoretical estimation of the number of nodes still assumes a full binary tree to derive the upper bound O(2n)O(2^n).

Example

Then, the instructor adds an important distinction: because the base case is f(1)=f(2)=1f(1)=f(2)=1, there is no need to fully expand the entire tree when drawing; yet for theoretical analysis, we still estimate the scale as 'this many nodes'. In other words, the ellipses shown in the diagram are merely drawing simplifications and do not alter the object of complexity analysis.

Common misconceptions

  • Seeing that the full tree is not drawn in the diagram, one might mistakenly think the theoretical scale of recursive calls also decreases.
  • Assuming that the base case changes the asymptotic upper bound of the time complexity.

Watch the explanation

2:37 – 2:50Watch this moment ↗

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