I came on this sequence of generating functions on trying to count lambda terms (in lambda calculus)

$T^{\langle m+1\rangle} = \frac{T^{\langle m \rangle}(z)}{z} - (T^{\langle m\rangle}(z))^2$

with

$T^{\langle m\rangle}(0) = 0.$

What can be said about these functions?

I would be happy to have an expression for $T^{\langle 0\rangle}(z)$ and say something about the asymptotic behavior of the coefficients $T_{n,m}$ for

$T^{\langle m\rangle}(z) = \sum_{n=0}^{\infty}\ T_{n,m} z^n.$