. This paper provides a characterization of the storage needs of a quadtree when used as an index to access large volumes of 2--dimensional data. It is shown that the page occupancy for data in random order approaches 33%. A precise mathematical analysis that involves a modicum of hypergeometric functions and dilogarithms, together with some computer algebra is presented. A brief survey of the analysis of storage usage in tree structures is included. The 33% ratio for quadtrees is to be compared to the figures for binary search trees (50%), tries (69%), and quadtries (46%). Computing Reviews Classification: E.1 [Data Structures], E.2 [Data Storage Representations], F.2.2 [Nonnumerical Algorithms and Problems], G.2.1 [Combinatorics] The research of this author was done while visiting INRIA, Rocquencourt, France under support from the Ministry of Education of Japanese Government. y Work of this author was supported in part by the Basic Research Action of the E.C. under contract No. 3..