OEIS A000094: Number of trees with diameter four

In this episode we unpack A000094, the count of unlabeled trees whose diameter is four. We start by clarifying what 'trees' and 'diameter' mean in graph theory, and review why you need at least five vertices to realize diameter four. Then we explore the first terms and what they count, and reveal two striking partition-based interpretations of the same sequence: (1) the number of partitions of N−1 into at least two parts of size at least two, and (2) the number of partitions of N−1 where the largest part is at least two larger than the smallest. We explain how these two seemingly different rules yield the same sequence, and show how the partition numbers p(n) come into play via A(n+1) = p(n) − n for n > 0. We close with real-world angles from computer science and optimization, and tease a second installment that digs even deeper into the connections between trees and partitions.

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