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 (2d−3.5)e+O(1/d).