I can't imagine any application. :-) The fascination is more that of explanation: if you treat T = 1 + T^2 as an algebraic equation in the complex numbers, you

I can't imagine any application. :-) The fascination is more that of explanation: if you treat T = 1 + T^2 as an algebraic equation in the complex numbers, you can get a lot of expressions that are meaningless: it's hard to make sense of T = 1/2 ± i√3/2 or T^3 = -1 or even T^6 = 1 (there is certainly no bijection sending six trees to the empty tree). But then, T^7 = T is suddenly very much meaningful and we can turn it into an explicit description. What's going on? Which equations are "meaningful" and which are not? Blass's paper gives an explanation: an equation is meaningful if you can derive it by working in (something like) N[T]/(T = 1 + T^2) (with some more conditions). Baez's discussion (linked above) puts it in terms of "categorification", and is also interesting.