Fractality is an important feature prevalent in real networks. However, a fundamental problem regarding the relationship between the fractal box dimension \(d_\textrm{b}\) and the cluster dimension \(d_\textrm{c}\) has remained unsolved. In this study, we develop an algorithm that generates the small-world scale-free fractal tree graph and perform a thorough exploration of the relationship between \(d_\textrm{b}\) and \(d_\textrm{c}\) using model networks generated by this algorithm. We derive an explicit relation that relates \(d_\textrm{b}\) to \(d_\textrm{c}\) in terms of the mean distance and diameter of networks. We show the validity of this relation through the investigation of the small-world scale-free fractal tree graph using a scale transformation that enables a direct comparison between the mean box and cluster sizes used to define the fractal dimensions. Furthermore, we show that the same analysis can be applied to real networks that exhibit fractal properties.