A vertex-irregular total k-labelling λ : V (G) ∪ E(G) −→ {1, 2, ..., k} of a graph G is a labelling of vertices and edges of G in such a way that for any different vertices x and y, their weights wt(x) and wt(y) are distinct. The weight wt(x) of a vertex x is the sum of the label of x and the labels of all edges incident with x. The minimum k for which a graph G has a vertex-irregular total k-labelling is called the total vertex irregularity strength of G, denoted by tvs(G). In this paper, we

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... termined the total vertex irregularity strengths of all quadtrees and all banana trees.