I am looking for an example that demonstrates the necessity of either subtour-elimination or of connectivity constraints in the LP formulation of the MST

In the internet I only could find the LP formulations, but no motivation for the constraints.

The need for topological constraints seems contradictory to me, because demanding the number of edges to be $n-1$, together with the requirement, that at least one edge be adjacent to each vertex, suffice for a connected graph of $n$ vertices to ensure connectivity and a tree topology. Fractional solutions seem also counter-intuitive.

So, where does the need for topoligical constraints in the LP formulation of the MST problem actually come from, resp. when does it arise?