# Distinct Minimum Weight Spanning Trees

I am trying to find the total number of distinct minimum weight spanning trees(MWST) in a simple, undirected, unlabeled and weighted graph but I am confused whether should I have to consider Isomorphic trees(if it is present) as a single minimum weight spanning tree or not while counting the distinct MWSTs because graph is unlabeled.

For example, If I have a graph as :

Now, I have to find the Total number of distinct minimum weight spanning trees(MWST) in the above weighted graph.

I got total $$3\times2 =6$$ possibilities of MWSTs but I also found the possibility of Isomorphic trees in these total $$6$$ MWSTs because graph is unlabeled. For example,the below $$2$$ MWSTs are Isomorphic in nature for the above given graph.

Similarly , If I have a simple, weighted, undirected and unlabeled graph as :

Now, I got total $$2^{2} \times 2^{4} = 64$$ MWSTs but If I consider Isomorphism property in a graph then I am getting 2 Isomorphic Trees as :

So, If I consider the possibility of many Isomorphic trees like above in the given graph then total number of distinct MWSTs will be less than $$6$$ and $$64$$ in the above 2 cases respectively. But I am not sure whether it is correct or not.

So, My doubt is :- While counting the total number of distinct MWSTs in an unlabeled weighted graph, Should I have to consider Isomorphic MWSTs as a single minimum weight spanning tree or not ?

Let me proceed with another natural and interesting approach according to the conditions. How can we count distinct distinct MSTs of a given unlabelled weighted graph $$G$$? Here is one reasonable procedure. Initially we have counted 0 MST. Now someone will present to us the MSTs one at a time together with $$G$$ to us. Every information that is available should be presented to us accurately, including whether two vertices are connected and what is the weight of each edge and which edge is selected as part of the current MST. How these information will be presented can be arbitrary. Vertices can be labelled arbitrarily. The order of edges presented can be arbitrary. If drawn graphically, the shape of each edge and the location of the each vertex can be arbitrary. Etc. Whenever we can be confident that we are seeing a new MST, we will increase the count by 1. Using this approach, which is probably what you meant by considering "isomorphic MWSTs as a single minimum weight spanning tree", we will count 3 MSTs for the first graph and 20 MSTs for the second graph.