We explore A000342, the OEIS entry that counts n-node rooted trees of exact height 5. Height here means the longest path from the root to a leaf is exactly 5 edges, which forces at least 6 nodes. So the early terms start 0, 0, 0, 1, 5, 19, 61, illustrating how the height constraint shapes the counts compared with unconstrained rooted trees. We’ll discuss the counting machinery—recurrences, generating functions, and dynamic programming—and how these counts relate to other related problems in combinatorics and computer science. Join us as we connect theory to data structures and networks, and reflect on how a small rule tweak can dramatically change the enumeration in the OEIS web of interlinked sequences.

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