{\sum } \psi = c
    \Psi_i^z,$$ where $\Psi_i=1+\lambda\xi_i$. As we have shown in our previous work, [@2011] the derived approximation for the partial partition function is equally applicable to the Cayley tree of $x$ generations capturing a chain connected to its $z^{\prime}=2(z-1)+(x-2)(z-2)$ nearest neighbor sites.

Determining the contribution to GPT of the Cayley tree with the occupied central site is considerably more mathematically demanding problem. The key element of the problem is to estimate the conditional probabilities of the occupancy of the nearest neighbor sites by other segments of the same chain molecule. In order to rationalize the above correlations among the occupancies of the aggregate sites imposed by the connectivity of the chain, we define the partial partition functions (PPF) $\chi _{k}(h)$. This PPF describes configurations of a part of the chain consisting of $k$ bonds extending outside a first generation sites of the Cayley tree and $i$ being the number of layers that separate the selected layer from the adsorbing surface. In terms of functions $\chi _{k,i}$, the contribution to the partition function coming from the aggregates with the occupied central site can be written [@2011] as $$\label{small_agg_GPF}
Z_i^{1}= c z \Psi _{x,i} ^{z-1} \lambda^{2} \eta \chi _{x-1,i} 