Cayley's formula from middle school math

The note contains a short elementary proof of Cayley’s formula for labeled trees.

💡 Research Summary

The paper titled “Cayley’s formula from middle school math” claims to give an elementary proof of the classical Cayley formula, which states that there are exactly (n^{,n-2}) labeled trees on (n) vertices. After a brief literature review that mentions the original determinant‑based proof by Borchardt, Cayley’s own polynomial argument, Prüfer’s bijection, and several modern expositions in the American Mathematical Monthly, the author announces a “middle‑school” approach that relies only on basic algebra and a simple case analysis.

The construction begins by drawing a path between vertex 1 and vertex 2. If the path contains (k_1) vertices (so there are (k_1-2) interior vertices), the number of ways to assign labels to those interior vertices is written as the product ((n-2)(n-3)\dots (n-k_1+1)). If (k_1=2) the product is empty and contributes a factor of 1.

Next a second path is added. This path must connect the smallest‑labelled vertex that has not yet been used (call it (x)) to some vertex on the already drawn structure. There are (k_1) possible attachment points, giving a factor (k_1). If the second path introduces (k_2-k_1) new vertices, the labeling choices for those new vertices contribute the factor ((n-k_1-1)(n-k_1-2)\dots (n-k_2+1)).

The process repeats: the (i)‑th path attaches to the current tree at one of the (k_{i-1}) existing vertices, adds (k_i-k_{i-1}) new vertices, and contributes the factor

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