This paper presents relations between hierarchic and parallel implementations of multirate filter banks (MRBs). An MRB consists of a set of filters that produce several reduced sampling-rate, or spatial scale, versions of a signal. Because image properties exist at all spatial scales, and different significance is placed on each of these scales, MRBs are very useful for image coding and analysis. Hierarchic processors, in which outputs are computed with a cascade of filters acting on a signal, have been shown in some special cases to implement MRBs more efficiently than parallel processors. However, no general theory for the hierarchic implementation of MRBs exists. We derive conversions between parallel multirate filter banks (PMRBs) and hierarchic multirate filter bank (HMRBs). The theory provides closed-form equations for finding the PMRB equivalent to a given HMRB provided a "commutation condition" (subsection 2.2.2) on the decimation and interpolation coefficients is satisfied. Closed-form solutions for the HMRB equivalent to a given PMRB can be found provided both the commutation condition and a "frequency-preservation condition" (subsection 2.3) on the PMRB filters are met. MRBs that do not satisfy the frequency preservation condition can not be implemented with an HMRB. We use a two-dimensional HMRB that allows arbitrary, non-decreasing rational-number reductions in sampling rate between successive outputs, arbitrary LTI filtering between outputs and low-pass filtering to prevent aliasing from the sampling-rate changes. Finally, we consider the special case of scaled Gaussian filters.