High girth augmented trees are huge

Let G be a graph consisting of a complete binary tree of depth h together with one back edge leading from each leaf to one of its ancestors, and suppose that the girth of G exceeds g. Let h=h(g) be the minimum possible depth of such a graph. The existence of such graphs, for arbitrarily large g, is proved in [2], where it is shown that h(g) is at most some version of the Ackermann function. Here we show that this is tight and the growth of h(g) is indeed Ackermannian.