We introduce some new symmetric tensor categories based on the combinatorics of trees: a discrete family \(\mathcal {D}(n)\) , for \(n \ge 3\) an integer, and a continuous family \(\mathcal {C}(t)\) , for \(t \ne 1\) a complex number. The construction is based on the general oligomorphic theory of Harman–Snowden, but relies on two non-trivial results we establish. The first determines the measures for the class of trees, and the second is a semi-simplicity theorem. These categories have some notable properties: for instance, \(\mathcal {C}(t)\) is the first example of a 1-parameter family of pre-Tannakian categories of superexponential growth that cannot be obtained by interpolating categories of moderate growth.
