Structure of spanning trees on the two-dimensional Sierpinski gasket

Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage n is a non-negative integer. For any given vertex x of SG(n), we derive rigorously the probability distribution of the degree j ε f1; 2; 3; 4g at the vertex and its value in the infinite n limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree j. The corresponding limiting distribution Φ_j gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as Φ₁ = 10957=40464, Φ₂ = 6626035=13636368, Φ₃ = 2943139=13636368, Φ₄ = 124895=4545456.