Trees with non-regular fractal boundary

For any given 0 < α < β < ∞, we construct a tree such that under tree metric, the Hausdorff dimension of the corresponding boundary is α, but both the Packing dimension and the boxing dimension are β. Applying the connection between tree and iterated functions system, nonregular fractal sets on real line are constructed. Moreover, the method adopted in our paper is different from those which have been used before for constructing non-regular fractal set in general metric space.