Proportion of spanning trees in a network in a social media messaging context

Consider a graph, such as the following:

I'm considering a model of message propagation (e.g. re-tweeting) in this network, starting from a root node (e.g. the node 1 in the lower-left). I'm modelling the message propagation in terms of trees rooted at the message source, where a node v is a parent to another node w if w first hears the message from v.

• The message propagates outward from 1, in a tree which "grows" in an iterative process. In this process, nodes which have been reached are either branching nodes (i.e. are the parent of some other node), live leaves (nodes which may become branch nodes), or dead leaves.
• Initially, all the neighbors of node 1 are children of 1, and are live leaves.

• The message propagates by iterations, as follows. We consider an arbitrary ordering L = (ℓ₁, ..., ℓ_n) of the live leaves at the beginning of the iteration. For each 1 ≤ j ≤ n, we do the following:
  

  1. Decide whether the node ℓ_j dies (doesn't propagate the message) or becomes a branch node (propagates the message). If all of the nodes adjacent to ℓ_j are already in the tree, then it dies by default.
  2. If ℓ_j becomes a branch node, we attach every neighbor v of ℓ_j which is not already in the tree to ℓ_j, as a live leaf node for the following iteration.
     

  After iterating through the elements of L, we proceed to the next iteration.

• If there are no more live leaves in the tree, we stop.

For the graph above, here are the trees that may be generated by this process:

I'm interested in considering how many of the trees which can be generated by this process are spanning trees, i.e. contain every node in the graph. Is there any formula to determine the ratio of the spanning trees to the total number of such trees?

N.B. The construction above is similar to a Galton-Watson process. However, there isn't meant to be a probabilistic model underlying the growth; the above is meant only to implicity describe a recursive process to recognise whether a subtree in the graph is valid in my model. I've added the probability tag just in case there is a useful approach from that direction.

Thanks!