OEIS A000107: Pointed rooted trees

What is A000107? It counts rooted trees with exactly one distinguished (labeled) vertex — the pointed or vertebrate rooted tree. The early terms are 0, 1, 2, 5, 13, 35, and the numbers grow rapidly as you add nodes. The combinatorial meaning is: pick a rooted tree and mark one node. Its ordinary generating function is A000081(X) / (1 - A000081(X)), linking it to the classic sequence of unlabeled rooted trees. Beyond counting, there are fascinating connections to circle arrangements in the plane studied by Mathar and to asymptotic growth formulas. Pointed rooted trees also surface in computer science (think file hierarchies) and biology (branching patterns in natural systems), showing how a single labeled node unlocks a rich structure. Join us as we explore the depth and beauty of this OEIS entry.

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