NumberOfSpanningTrees command not working correctly

The Combinatorica package is now considered obsolete and not compatible with Mathematica's built-in Graph theory functionality. The function CycleGraph, which you are using, is a built-in function and is not understood by the Combinatorica package. You have to replace CycleGraph[i] simply by Cycle[i], which is the corresponding object in Combinatorica. You can confirm this that by looking at (for example) CycleGraph[5] and ShowGraph[Cycle[5]]:

GraphicsRow[{CycleGraph[5] , ShowGraph[Cycle[5]]}]

Actually, the Combinatorica package still has a lot of functionality that has nothing equivalent in the Kernel (and it is not clear if this will ever change completely). On the other hand it is essentially undocumented (unless you buy the book by Skiena and Pemmaraju) and the quality of Mathematica programming is not great (there are also bugs that never seem to get fixed).

Without using the Combinatorica package, you can use

 Table[{i, GraphData[{"Cycle", i}, "SpanningTreeCount"] == i}, 
{i, 3, 20}] // TableForm

to get

 {{3, True}, {4, True}, {5, True}, {6, True}, {7, True}, {8, True}, 
 {9, True}, {10, True}, {11, True}, {12, True}, {13, True}, {14, True}, 
 {15, True}, {16, True}, {17, True}, {18, True}, {19, True}, {20, True}}

This does not work for i>20 since GraphData contains named cycle graphs {"Cycle", i} for i<=20.