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

Abstract: 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 $\vec{\beta}$-labeling (oriented beta-labeling). Our proof employs the polynomial method by identifying trees as functions in the transformation monoid $\mathbb{Z}_n^{\mathbb{Z}_n}$. A proof of the graceful tree conjecture (1967) follows as an immediate consequence of the current result. Finally, we introduce additional algebraic properties derived from the decomposition results.