Measures on Cameron's treelike classes and applications to tensor categories

Measures on Fra\"iss\'e classes are a key input in the Harman--Snowden (2022) construction of tensor categories. Treelike Fra\"iss\'e classes provide a particularly tractable source of examples. In this paper, we complete the classification of measures on Cameron's elementary treelike classes. In particular, for the class $\partial \mathfrak{T}_3(n)$ of node-colored rooted binary tree structures with $n$ colors, we classify measures by an explicit bijection with directed rooted trees edge-labeled by $\{1, \dots, n\}$ with a distinguished vertex, yielding $(2n+2)^n$ distinct $\mathbb{Z}\left[\frac{1}{2}\right]$-valued measures. For each $n \geq 1$, we use a family of measures $\mu_n^I$ and their supports $\partial \mathfrak{T}_3(n)^{\mathrm{ord}}_I$ (where $I \subseteq \{1, \dots, n\}$) to construct the Karoubi envelopes $\mathbf{Rep}(\partial \mathfrak{T}_3(n)^{\mathrm{ord}}_I;\mu^I_n)$, producing infinite families of semisimple tensor categories with superexponential growth that cannot be obtained via Deligne's interpolation of representation categories. We also prove the nonexistence of measures on the $n$-colored tree class $C_n\mathfrak{T}$ for $n \geq 2$ and the labeled tree class $L \mathfrak{T}$, extending Snowden's results for uncolored trees.