By building the full cascade system for each class of operator, space and spatial-frequency respectively, a single tree structured decomposition is obtained. The single trees can be grown to arbitrary depth, 
, where N is the length of the signal. A single tree built from filter bank units implements the tree-structured wavelet transform, while the single tree built from the segmentation operators implements the quad-tree segmentation.
The tree-structured wavelet transform, or wavelet packet tree, is constructed by recursively passing the output of each analysis filter channel through another instance of the filter bank. This is illustrated in Figure 6(a) and 7(a). The wavelet packet tree has the advantage of attaining the complete hierarchy of segmentations in frequency. By selectively choosing the frequency segments, the signal can usually be represented at lower storage cost than the original signal. The wavelet packet compression produces its coding gain by choosing the frequency segmentation that best captures a frequency spectrum that is not flat. However, as mentioned, the frequency spectrum for the signal as a whole is analyzed. Wavelet packets cannot compensate for or take advantage of signal nonstationarity.
The full quad-tree spatial segmentation is obtained by cascading the segmentation operations such that each segmentation output is again segmented completely. This is illustrated in Figure 6(b) and 7(b). The quad-tree segmentation can be used to best compensate for image non-stationarity. By selectively choosing from the hierarchy of spatial segments, the image can be represented at lower cost. For images quad-tree spatial segmentation does not have the same energy packing ability as the filter bank. However, many images have largely diverse content in different spatial regions, and the independent treatment of regions can provide some gain.
Figure 7: Using notation above, (a) tree-structured wavelet transform or wavelet packet tree, (b) hierarchical segmentation.