  We generalize a tensor-network algorithm to study thermodynamic properties of
self-similar spin lattices constructed on a square-lattice frame with two types
of couplings, $J_{1}^{}$ and $J_{2}^{}$, chosen to transform a regular square
lattice ($J_{1}^{} = J_{2}^{}$) onto a fractal lattice if decreasing $J_{2}^{}$
to zero (the fractal fully reconstructs when $J_{2}^{} = 0$). We modified the
Higher-Order Tensor Renormalization Group (HOTRG) algorithm for this purpose.
Single-site measurements are performed by means of so-called impurity tensors.
So far, only a single local tensor and uniform extension-contraction relations
have been considered in HOTRG. We introduce ten independent local tensors, each
being extended and contracted by fifteen different recursion relations. We
applied the Ising model to the $J_{1}^{}-J_{2}^{}$ planar fractal whose
Hausdorff dimension at $J_{2}^{} = 0$ is $d^{(H)} = \ln 12 / \ln 4 \approx
1.792$. The generalized tensor-network algorithm is applicable to a wide range
of fractal patterns and is suitable for models without translational
invariance.
