  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.
