Is it an irony of a high order to embellish an axe with the Tree of Life symbol, the Lebensbaum the Yggdrasil? Probably from one point of view or the other one the answer could be yes.
On the face of it though, that is to say if we are to put ourself in the mind of the one who possibly made/forged, or was it the one who placed the order for? such an axe, (and we are going on then about that one who was actually breathing maybe even as late as say 1950 but which in theory might also have been 1450) no doubt the impuls was one of piety towards Jesus. But I always say that we can be better served by paying little or no attention to what motivates others. My point here is the true origin of the symbolic representation depicted on this broadaxe out of Austria is more universal, one might even locate it in the collective consciousness introduced to us by that other Austrian C. Jung. Do you find the coincidence convincing enough? The smid, the priest, the zimmermann, and the philosopher all finding a unity in the expression so tangible, so timeless and self-reproducing as the tool taken in hand.