Fundamental group of the sphere via triangulation

I know that the fundamental group of the sphere is zero, i.e. $pi(S^2)=0$

I want to show this by triangulation, i.e:

1. Triangulate the sphere
2. Draw maximal tree
3. Draw maximal contractable subspace
4. Consider generators on remaining 1-simplices

Here is what happened:

I drew the following triangulation:

I then proceeded to draw the maximal tree. But to include all vertices, and due to the imposed identifications, I found that this was just the boundary, so not a tree

So can we just conclude that since we cannot carry out the process, the fundamental group is zero? I was wondering how to do this formally -perhaps I am missing a step?

Many thanks