OEIS A00014: Series-reduced trees

In this Deep Dive we explore the idea of series-reduced trees—a family of rooted trees with a tidy rule: every internal node either has no children (a leaf) or at least two children, with no single middle-child case. We connect this structure to the counting sequence A00014, discuss how the number of distinct trees grows as vertices are added (including moments where multiple trees exist for a given size), and thread the discussion through the broader language of graph theory. We’ll touch on the Beattie–McKay catalog of all unique series-reduced trees up to 22 nodes, the historical note about a counting formula with a small omission corrected by Wolf Dieter Lange, and related connections to classic diagram-counting puzzles popularized in both mathematics and cinema. Expect visuals, intuition, and a map of how a single sequence can illuminate seemingly distant corners of mathematics.

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