A Note On Vertex Distinguishing Edge colorings of Trees

A proper edge coloring of a simple graph $G$ is called a vertex distinguishing edge coloring (vdec) if for any two distinct vertices $u$ and $v$ of $G$, the set of the colors assigned to the edges incident to $u$ differs from the set of the colors assigned to the edges incident to $v$. The minimum number of colors required for all vdecs of $G$ is denoted by $\chi\,'_s(G)$ called the vdec chromatic number of $G$. Let $n_d(G)$ denote the number of vertices of degree $d$ in $G$. In this note, we show that a tree $T$ with $n_2(T)\leq n_1(T)$ holds $\chi\,'_s(T)=n_1(T)+1$ if its diameter $D(T)=3$ or one of two particular trees with $D(T) =4$, and $\chi\,'_s(T)=n_1(T)$ otherwise; furthermore $\chi\,'_{es}(T)=\chi\,'_s(T)$ when $|E(T)|\leq 2(n_1(T)+1)$, where $\chi\,'_{es}(T)$ is the equitable vdec chromatic number of $T$.