We have used the fractal nature of the geometry in obtaining approximate design equations for dipole antennas with Hilbert curve geometry. We have also reported that in the case of Koch curves, the fractal dimension can be varied by changing the indentation angle, and the resonant frequencies of the resultant antenna follow a close relation with the fractal dimension. It may be recalled that fractal dimension is an important characteristic of fractal geometries. However this is not a unique description for the geometry, but rather identifies a group of geometries with similar nature. Hence a first step in the utilization of fractal properties in antenna design should involve the dimension of the geometry. This paper is a step towards justifying the link between dipole antenna characteristics and the fractal properties of the geometry. We have found in cases where the appearance of the geometry is perturbed in such a way that its similarity dimension is varied, the multiband characteristics is affected, where as other perturbations leave this rather unaffected.. This gives a further proof that antenna characteristics can in fact be linked to the fractal dimension of such geometries. To enable two such variations we have used designed dipole antennas using fractal tree geometry. To distinguish this geometry from the previously reported tertiary tree geometries we refer to this as fractal binary tree.