Every tree on $n$ edges decomposes $K_{nx,nx}$ and $K_{2nx+1}$

We prove that every tree on $n$ edges decomposes $K_{nx,nx}$ and $K_{2nx + 1}$ for all positive integers $x$. The said decompositions are obtained by proving that every tree admits a $\vecβ$-labeling (oriented beta-labeling). Our proof employs the polynomial method by identifying trees as functions