  The number of spanning trees of a graph is an important invariant related to
topological and dynamic properties of the graph, such as its reliability,
communication aspects, synchronization, and so on. However, the practical
enumeration of spanning trees and the study of their properties remain a
challenge, particularly for large networks. In this paper, we study the num-
ber and degree distribution of the spanning trees in the Hanoi graph. We first
establish recursion relations between the number of spanning trees and other
spanning subgraphs of the Hanoi graph, from which we find an exact analytical
expression for the number of spanning trees of the n-disc Hanoi graph. This
result allows the calculation of the spanning tree entropy which is then
compared with those for other graphs with the same average degree. Then, we
introduce a vertex labeling which allows to find, for each vertex of the graph,
its degree distribution among all possible spanning trees.
