We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in An operad of non-commutative independences defined by trees (Dissertationes Mathematicae, 2020, doi:10.4064/dm797-6-2020), which generalize the free, boolean, monotone, and orthogonal convolutions. In particular, for each rooted subtree $mathcal{T}$ of the $N$-regular tree (with vertices labeled by alternating strings), we define the convolution $boxplus_{mathcal{T}}(mu_1,dots,mu_N)$ for arbitrary probability measures $mu_1$, ..., $mu_N$ on $mathbb{R}$ using a certain fixed-point equation for the Cauchy transforms. The convolution operations respect the operad structure of the tree operad from doi:10.4064/dm797-6-2020. We prove a general limit theorem for iterated $mathcal{T}$-free convolution similar to Bercovici and Patas results in the free case in Stable laws and domains of attraction in free probability (Annals of Mathematics, 1999, doi:10.2307/121080), and we deduce limit theorems for measures in the domain of attraction of each of the classical stable laws.
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