The Growth Rate of High-Dimensional Tree Polycubes

A d-dimensional polycube is a ( d − 1 ) -face-connected set of cubes in d dimensions. Fixed polycubes are considered distinct if they differ in shape or orientation. The cube-adjacency graph of a tree polycube is a tree. In this paper we investigate the asymptotic growth rate of d-dimensional tree polycubes, and estimate it at ( 2 d − 3.5 ) e + O ( 1 / d ) .