dependent quantities.

In order to rationalize the above derived contributions to the partition functions one must rely on some assumptions about the structure of the lattice surrounding the aggregate, as described by the fields $\xi$ and $\chi_k$. The model is appropriate for describing the considered system by the Cayley tree with the number of generations of brunches not larger than the number of polymer bonds, which takes into account local configurations. Non-local configurations are taken into account using the axillary fields $\xi$ and $\chi_k$ resolved by the equal probability condition [@Roberts; @Chang1939] that the occupancy state of the central site is equal to that of sites of the first generation sites of the Cayley tree. We will elaborate on that in what follows. The introduced partial partition function $\chi_k$ that describes the correction to the partition function beyond its dimeric part for the outside part of chain has been previously calculated for the homogeneous system [@2011] using the above tree structure in analogy with the homogeneous Bethe lattice theory [@Ryu-Gujrati1997]: $$\label{hom_chi}
    \chi_k = p\chi_{k-1},k\geq1,$$ where $p$ is the probability to have one extra bond extending beyond the center of the Cayley tree, $p = r_f \lambda \eta \Psi_x^{-1}\chi_1$ and $\chi_1 