Isomorphic Factorization of Trees of Maximum Degree Three

We prove that for any tree \(T\) of maximum degree three, there exists a subset \(S\) of \(E(T)\) with \(|S| = O(\log n)\) and a two-coloring of the edges of the forest \(T \setminus S\) such that the two monochromatic forests are isomorphic, where \(n\) is the number of vertices of \(T\) of degree three.