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The number of spanning trees of a graph is an important invariant related totopological and dynamic properties of the graph, such as its reliability,communication aspects, synchronization, and so on. However, the practicalenumeration of spanning trees and the study of their properties remain achallenge, particularly for large networks. In this paper, we study the num-ber and degree distribution of the spanning trees in the Hanoi graph. We firstestablish recursion relations between the number of spanning trees and otherspanning subgraphs of the Hanoi graph, from which we find an exact analyticalexpression for the number of spanning trees of the n-disc Hanoi graph. Thisresult allows the calculation of the spanning tree entropy which is thencompared with those for other graphs with the same average degree. Then, weintroduce a vertex labeling which allows to find, for each vertex of the graph,its degree distribution among all possible spanning trees.

The number and degree distribution of spanning trees in the Tower of Hanoi graph