The growth rate of high-dimensional tree polycubes

A d -dimensional polycube is a face-connected set of cubes in  d dimensions, where the faces are ( d − 1 ) dimensional. Fixed polycubes are distinct if they differ in shape or orientation. The cube-adjacency graph of a tree polycube does not have cycles. In this paper we estimate the asymptotic growth rate of d -dimensional tree polycubes at ( 2 d − 3.5 ) e + O ( 1 / d ) .