given by (\[inhom\_chi\]) by extending the recursive relation (\[hom\_chi\]) to the case $k = x$ to yield $$\label{field_h}
    1-h = (p^{2} \chi_{1}^{-1})^{1/(x+1)}.$$ In addition, we define $P_k(p,h) \equiv \chi_k(p,h)/\gamma^{1/2}$, where $\gamma$ is the normalization factor that is introduced to have the bulk properties far away from the surface, which is represented by $$\label{gamma_norm}
    \left(\sum_{k=0}^{x-1}P _{k} P _{x-k-1}\right)_b=x,$$ Using the latter definition in (\[small\_agg\_GPF\]), one finds that $$\label{small_agg_GPF1}
Z = c\Psi^{z} + c\frac{z}{2} r_{f}^{-1}  \lambda \Psi _{x} ^{z} p^{-2} \gamma \underset{k=0}{ \overset{x-1}{\sum }}P _{k} P _{x-k-1},$$

The above arguments about the connection between homogeneous and non-homogeneous partial partition functions can be generalized to to the recursive relations on the Cayley tree occupying 3 consecutive layers with the same ghost field $h$ depending on number of steps from the surface: $$\begin{aligned}
 \label{req_rl_P}
P _{0} &=& (1-h)\chi_0/\gamma^{1/2}, \nonumber \\
P _{1} &=& (1-h)p (\lambda_0 P_0 + \lambda_1 P_{0}^{(-)} + \lambda_1 P_0^{(+)}), \nonumber \\
&&\vdots \\
P _{k} &=& (1-h)p (\lambda_0 P_{k-1} + \lambda_1 P_{k-1}^{(-)} + \lambda_1 P_{k}^{(+)}),  \nonumber\end{aligned}$$ Here, the coupling constants $\lambda_1$ and $\lambda_0$ weight contributions from the three neighboring layers and $\lambda_0 = (1-\lambda_1)/2$ is the fraction of the average coordination number $r_f$ to come back to the same level at 