Subtraction Trees Express Factorial n! as Function of Polynomial x_n

Differentiating $x^n$ by $n$ times gives $n!$, but what is the explicit function that connects the two? This article offers the unique insight on how the polynomial $x^n$ is found inside the series that expresses the factorial $n!$, i.e. $n!=f(x^n)$. Subtraction trees are the mathematical mechanism used to establish this connection. The process is here applied to powers $n=2 \to 6$, but this can be extended to any power $n$. Proofs using the mathematical method of induction are provided for each power, resulting in the respective function expression. Moreover, reworking this new function $n!=f(x^n)$ enable the determination of all the coefficients $^nC_k$ in a row $n$ of the Pascal's triangle (a worked example is provided). A Matlab/Octave program to compute this is enclosed for practical classroom activities.