I cannot see a clear question here, and so this is certainly not an answer. But perhaps you could clarify the question via an explicit example. As I understand it, if you restricted attention to triangulated surfaces homemorphic to a sphere (which I know is not your interest), your cutting would produce what is called a net, or an unfolding. You are looking at the dual of the net, and want either bushy trees or Hamiltonian paths. There are exactly 43,380 distinct nets for the icosahedron. Left below is an unfolding with a bushy dual tree; on the right an unfolding whose dual is a Hamiltonian path. <br /> ![Icosa Nets][1] <br /> The only points I want to make with this example are: (a) There are _many_ spanning trees (exponential in the number of triangles), and (b) among them you can probably find spanning trees of any desired shape. [1]: https://i.sstatic.net/vX4qv.jpg