= \Psi_x^{z-1}\Psi^{-(z-1)}$; with the power $(z-2)$ applied to $\Psi_x$ due to having $z-2$ local gauche conformations out of $z-1$ possible for a middle segment located anywhere except the center of the Cayley tree, and the power $-(z-1)$ is applied to $\Psi$ that is being deduced [@2011] from the dimer theory[@Chang1939] by taking the limit $x=2$. For the purposes of the present work, this relation is to be extended to the case of non-uniform systems. We observe that the definition $$\label{chi_0}
    \chi_0 \equiv p^{-1} \chi_{1}$$ allows to incorporate both terms in (\[small\_agg\_GPF\]) to the same sum with extended limits. Formally, $\chi_0$ corrects for the absence of the bond in the end point of the polymer chain.

In order to incorporate the above mentioned difference between the behavior of the end and middle segments near the adsorbing surface into the above lattice model one has to resort to the standard method of percolation theory [@Stinchcombe; @Reynolds] on the Cayley tree that describes critical phenomena by making use the concept of “ghost site” (GS). This construction is illustrated in Fig. \[Fig1\]b. As can be seen GS is simultaneously connected to all sites of the tree with probability $h$ per site. This concept can 