Summary: We study certain notions of \(N\)-ary non-commutative independence, which generalize free, Boolean, and monotone independence. For every rooted subtree \(\mathcal{T}\) of an \(N\)-regular rooted tree, we define the \(\mathcal{T}\)-free product of \(N\) non-commutative probability spaces and the \(\mathcal{T}\)-free additive convolution of \(N\) non-commutative laws.
These \(N\)-ary additive convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the operadic framework, the proof of convolution identities such as \(\mu\boxplus\nu = \mu \vartriangleright (\nu\,\square\!\!\!\!\!\vdash \!\mu) \) can be reduced to combinatorial manipulations of trees. In particular, we obtain a decomposition of the \(\mathcal{T} \)-free convolution into iterated Boolean and orthogonal convolutions, which generalizes work of R. Lenczewski [J. Funct. Anal. 246, No. 2, 330–365 (2007; Zbl 1129.46055)].
We also develop a theory of \(\mathcal{T} \)-free independence that closely parallels the free, Boolean, and monotone cases, provided that the root vertex has more than one neighbor. This includes combinatorial moment formulas, cumulants, a central limit theorem, and classification of distributions that are infinitely divisible with bounded support. In particular, we study the case where the root vertex of \(\mathcal{T}\) has \(n\) children and each other vertex has \(d\) children, and we relate the \(\mathcal{T} \)-free convolution powers to free and Boolean convolution powers and the Belinschi-Nica semigroup.