Summary: In [Discrete Math. 310, No. 21, 3043--3048 (2010; Zbl 1208.05014)], \textit{Nurdin} et al. conjectured that the total vertex irregularity strength of any tree \(T\) is determined only by the number of vertices of degrees 1, 2 and 3 in \(T\). This paper will confirm this conjecture by considering all trees with maximum degree five. Furthermore, we also characterize all such trees having the total vertex irregularity strength either \(t_1\), \(t_2\) or \(t_3\), where \(t_i = \lceil (1+\sum_{j=1}^{i}n_j)/(i+1)\rceil\) and \(n_i\) is the number of vertices of degree \(i\).