  Spanning trees are relevant to various aspects of networks. Generally, the
number of spanning trees in a network can be obtained by computing a related
determinant of the Laplacian matrix of the network. However, for a large
generic network, evaluating the relevant determinant is computationally
intractable. In this paper, we develop a fairly generic technique for computing
determinants corresponding to self-similar networks, thereby providing a method
to determine the numbers of spanning trees in networks exhibiting
self-similarity. We describe the computation process with a family of networks,
called $(x,y)$-flowers, which display rich behavior as observed in a large
variety of real systems. The enumeration of spanning trees is based on the
relationship between the determinants of submatrices of the Laplacian matrix
corresponding to the $(x,y)$-flowers at different generations and is devoid of
the direct laborious computation of determinants. Using the proposed method, we
derive analytically the exact number of spanning trees in the $(x,y)$-flowers,
on the basis of which we also obtain the entropies of the spanning trees in
these networks. Moreover, to illustrate the universality of our technique, we
apply it to some other self-similar networks with distinct degree
distributions, and obtain explicit solutions to the numbers of spanning trees
and their entropies. Finally, we compare our results for networks with the same
average degree but different structural properties, such as degree distribution
and fractal dimension, and uncover the effect of these topological features on
the number of spanning trees.
