These claims, at least in their simple wording, are not true. In this case, the top of the tree is the top of the trunk. We can get examples like this even with small theta values. More analysis is needed here. However, they offer no proof of this.
More analysis is needed here.
Clearly, the closed form can be written in many other ways. My equation has an obvious interpretation in the geometry of the tree, though its geometric derivation is less obvious. Finally, there is a lovely relation to the Moran generator equation, which is explainable in terms of the geometric derivations of the two equations. Now separate the first two terms from the first sum, noting that the second term is zero, then combine the two sums, term by term. It looks nice, it is short and sweet.