The growth rate of high-dimensional tree polycubes

A d-dimensional polycube is a face-connected set of cubes in  ddimensions, 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 (2d−3.5)e+O(1/d).