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, $\Lambda$, of the partition function $Z_N=N^{\theta}\Lambda^N$ 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 $\ln \Lambda = \lim_{N\to\infty} \ln Z_N / N$ with $\Lambda= 2(\sqrt{2}+1) \approx 4.8284$.