Scaling laws of chemical and Euclidean distances in critical percolation trees

We study the scaling behavior of chemical distances in spanning trees of critical site percolation clusters. Precisely, we analyze spanning trees constructed using four different algorithms: minimal path trees (MPTs), depth-first search (DFS), right-first search (RFS), and minimum spanning trees (MSTs). In each case, the chemical distance ℓ is not defined in the original percolation graph, but rather on the specific spanning tree generated by the chosen traversal rule. We analyze the conditional probability distribution P_{r}(r|ℓ), where r is the Euclidean distance given a chemical distance ℓ, and show that it follows a universal scaling form. Using scaling arguments and Bayes' theorem, we derive and numerically verify general relations between the characteristic exponents that govern these scaling laws. Each spanning-tree construction reveals the geometry of a different self-similar structure within the percolation cluster: MPTs yield the fractal dimension of the minimal path, MSTs recover the optimal path dimension, and DFS and RFS give the fractal dimension of the external hull in two dimensions. In three dimensions, the DFS tree reveals a previously unreported exponent, θ_{1}=1.575, suggesting a new universality class for this type of self-similar structure. These results establish a general framework for extracting fractal dimensions of paths embedded in disordered media via spanning-tree-induced chemical distances.