A k-(2, 1)-total labelling of a graph G is a mapping f:V(G)∪E(G)→{0,1,…,k}\documentclass[12pt]{minimal}
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\begin{document}$$f: V(G)\cup E(G)\rightarrow \{0,1,\ldots ,k\}$$\end{document} such that adjacent vertices or adjacent edges receive distinct labels, and a vertex and its incident edges receive labels that differ in absolute value by at least 2. The (2, 1)-total number, denoted λ2t(G)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _2^t(G)$$\end{document}, is the minimum k such that G has a k-(2, 1)-total labelling. Let T be a tree with maximum degree Δ≥7\documentclass[12pt]{minimal}
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\begin{document}$$\Delta \ge 7$$\end{document}. A vertex v∈V(T)\documentclass[12pt]{minimal}
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\begin{document}$$v\in V(T)$$\end{document} is called major if d(v)=Δ\documentclass[12pt]{minimal}
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\begin{document}$$d(v)=\Delta $$\end{document}, minor if d(v)<Δ\documentclass[12pt]{minimal}
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\begin{document}$$d(v)<\Delta $$\end{document}, and saturated if v is major and is adjacent to exactly Δ-2\documentclass[12pt]{minimal}
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\begin{document}$$\Delta - 2$$\end{document} major vertices. It is known that Δ+1≤λ2t(T)≤Δ+2\documentclass[12pt]{minimal}
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\begin{document}$$\Delta + 1 \le \lambda _2^t(T)\le \Delta + 2$$\end{document}. In this paper, we prove that if every major vertex is adjacent to at most Δ-2\documentclass[12pt]{minimal}
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\begin{document}$$\Delta -2$$\end{document} major vertices, and every minor vertex is adjacent to at most three saturated vertices, then λ2t(T)=Δ+1\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _2^t(T) = \Delta + 1$$\end{document}. The result is best possible with respect to these required conditions.
