Partial match queries constitute the most basic type of associative queries in multidimensional data structures such as -d trees or quadtrees. Given a query where s of the coordinates are specified and are left unspecified (), a partial match search returns the subset of data points in the data structure that match the given query, that is, the data points such that whenever . There exists a wealth of results about the cost of partial match searches in many different multidimensional data structures, but most of these results deal with random queries. Only recently a few papers have begun to investigate the cost of partial match queries with a fixed query . This paper represents a new contribution in this direction, giving a detailed asymptotic estimate of the expected cost for a given fixed query . From previous results on the cost of partial matches with a fixed query and the ones presented here, a deeper understanding is emerging, uncovering the following functional shape for <(l.o.t. lower order terms, throughout this work) in many multidimensional data structures, which differ only in the exponent and the constant , both dependent on s and K, and, for some data structures, on the whole pattern of specified and unspecified coordinates in as well. Although it is tempting to conjecture that this functional shape is "universal", we have shown experimentally that it seems not to be true for a variant of -d trees called squarish -d trees.