OEIS A000235: Rooted trees of height exactly 3

In this episode we explore A000235, the number of labeled rooted trees on n nodes whose height is exactly 3. We recap what a rooted tree and its height mean, note why the first nonzero terms occur at n = 4 (a simple path) and how additional nodes can be attached without exceeding height 3, and look at the early values 0, 0, 0, 1, 3, 8, 18, … . We’ll touch on the main counting ideas: a Fibonacci-based convolution with the height-at-most-2 trees, an inclusion-exclusion formula involving partitions, and generating-function techniques. Plus a nod to the historical groundwork (Riordan, Sloan) and practical computation via Maple/Mathematica code, with context inside the broader OEIS table of counts by height.

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