Nonexistent Properties of Galton-Watson Trees

Let p be the probability for a Galton-Watson tree with offspring distribution Poisson(m) to contain an infinite binary tree starting at its root. This holds for the tree if and only if it holds for at least two of its child subtrees, yielding a recurrence satisfied by p. When m is sufficiently large, the recurrence has three solutions, the largest of which is the true probability. Do the other two solutions correspond to different properties that also hold if and only if they hold for at least two child subtrees?

The context for this question is in work by Moumanti Podder and Joel Spencer on first-order probabilities of Galton-Watson trees, which I'll describe in more detail. The solution is equivalent to proving that a certain recursive tree process is endogenous, in the terminology of Aldous and Bandyopadhyay. This is ongoing work with Podder and Spencer.