OEIS A000072: Number of labeled trees with n nodes

We explore Cayley’s formula a(n) = n^(n-2) and the surprising ubiquity of this count—from spanning trees of the complete graph K_n to parking functions, chip-firing configurations, cycle factorizations in the symmetric group, acyclic mappings, and even algebraic discriminants tied to roots of unity. This episode reveals how one simple sequence ties together broad strands of combinatorics, algebra, and number theory.

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