### Structure of spanning trees on the two-dimensional Sierpinski gasket

*Shu-Chiuan Chang, Lung-Chi Chen*

#### Abstract

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
∈{1,2,3,4} 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 φ_{1}=10957/40464, φ_{2}=6626035/13636368, φ_{3}=2943139/13636368, φ_{4}=124895/4545456.Full Text: PDF PostScript