Average number of facets per cell in tree-structured vector quantizer partitions

Upper and lower bounds are derived for the average number of facets per cell in the encoder partition of binary tree-structured vector quantizers. The achievability of the bounds is described as well. It is shown that the average number of facets per cell for unbalanced trees must lie asymptotically between three and four in R² , and each of these bounds can be achieved, whereas for higher dimensions it is shown that an arbitrarily large percentage of the cells can each have a linear number (in codebook size) of facets. Analogous results are also indicated for balanced trees