+ 2^{-1}
r_{f} c z
 \Psi _{x,i} ^{z-2} \lambda^3 \eta^2 \underset{k=1}{
\overset{x-2}{\sum }}\chi _{k,i}\chi _{x-k-1,i},$$ where $\Psi_{x,i} \equiv (1+\lambda \eta \xi_{i} )$. Here, the first term in the first term in the right hand side (r.h.s.) corresponds to the endpoint residing in the central site taking into account all $z$ possibilities for the first bond to extend from from the center of the Cayley tree. The second term takes into account the case when the central site is occupied by the middle segment with the coefficient $r_{f} = w_{f}(z-2)+1$ accounting for the finite stiffness of the chain, i.e. the latter weights $z-2$ out of $z-1$ possible configurations with the the Boltzmann weight $w_f$. The total GPF of a single layer separated from the adsorbing surface by $i$ lattice steps is given by a sum of the partition functions given by Eqs.(\[small\_agg\_GPF\]) and (\[GPF\_sm\_cluster\]). Note that in contrast to Eq.(\[GPF\_sm\_cluster\]), Eq.(\[small\_agg\_GPF\]) incorporates the parameter $\eta$ that describes the nearest-neighbor interactions between the segment placed into the central cite and the other segments that belong to the same aggregate. Note that the index $i$ is not included in the presentation for further convenience but assumed implicitly in what follows for all layer 