A new vertex distinguishing total coloring of trees

Let $ f $ be a proper total $ k $-coloring of a simple graph $ G $ from $ V(G)\cup E(G) $ to $ \{1, 2, \dots, k\} $, let $ C(u, f) $ be the set of the colors assigned to the edges incident with $ u $, and let $ n_d(G) $ and $ \Delta(G) $ denote the number of all vertices of degree $ d $ and the maximum degree in $ G $, respectively. We call $ f $ a (2)-vertex distinguishing total $ k $-coloring ($ k $-(2)-vdc for short) if $ C(u, f)\neq C(v, f) $ and $ C(u, f)\cup \{f(u)\}\neq C(v, f)\cup \{f(v)\} $ for distinct vertices $ u, v\in V(G) $. The minimum number $ k $ of colors required for which $ G $ admits a $ k $-(2)-vdc is denoted by $ \chi''_{2s}(G) $. In this paper, we show that a tree $ T $ with $ n_2(T)\leq n_1(T) $ has $ \chi''_{2s}(T) = n_1(T) $ if and only if $ T $ is not a tree with $ D(T) = 2, 3 $ or $ n_1(T) = \Delta(T) $, where $ D(T) $ is the diameter of tree $ T $.