be rationalized by introducing surface auxiliary field $h$ that describes the corrections to the probabilities of the configurations due to the presence of the surface. The presence of GS modifies the form of the bulk PPFs that can be written as $$\label{inhom_chi}
    \chi_k(p,h) = \chi_k(p)(1-h)^{k+1},$$ The case of the bulk PPF can be retrieved from (\[inhom\_chi\]) taking $h=0$. Non-zero $h$ corresponds to the situations where each site of the chain is connected to the GS which mimics polymer-surface interaction.

In order to determine $h$, it is necessary to consider the quantity $\chi_x(p,h)$ representing the probability to have simultaneously $x$ sites and bonds comprising the chain. The configuration described by $\chi_x(p,h)$ violates the above property of the Conventional Cayley tree of not having closed loops. The formation of such loop is solely due to the presence of the GS. For this special configuration at least one site of the Cayley tree occupied by the chain has to be connected to GS, which corresponds to $h=0$. Owing to the absence of the end-points in the closed loops $\chi_x(p,h)$ factorizes into $x$ probabilities $p$ of the bond formation. Thus simple relation $\chi_x(p,h)=p^{x}$ allows us to define $h$ as an analytical continuation of $\chi_k(p,h)$ 