Combinatorics of 3D directed animals on a simple cubic lattice

We provide combinatorial arguments based on a two-dimensional extension of a locally-free semigroup allowing us to compute the growth rate, $\mathrm{\Lambda }$, of the partition function $Z_N=N^{\theta }{\mathrm{\Lambda }}^N$ of the $N$-particle directed animals ($N\gg 1$) on a simple cubic lattice in a three-dimensional space. Establishing the bijection between the particular configuration of the lattice animal and a class of equivalences of words in the 2D projective locally-free semigroup, we find we find $\ln \mathrm{\Lambda }={\lim }_{N\to \mathrm{\infty }}\ln Z_N\mathrm{/}N$ with $\mathrm{\Lambda }=2(\sqrt{2}+1)\approx 4.8284$.