(Random) Trees of Intermediate Uniform Growth

For every sufficiently well-behaved function $g:\mathbb{R}_{\ge 0}\rightarrow\mathbb{R}_{\ge 0}$ that grows at least linearly and at most exponentially we construct a tree $T$ of uniform volume growth $g$, that is, $$C_1\cdot g(r/4)\le |B_{T}(v,r)| \le C_2\cdot g(4r),\quad\text{for all $r\ge 0$ and $v\in V(T)$},$$ where $B_{T}(v,r)$ denotes the ball of radius $r$ centered at a vertex $v$. In particular, this yields examples of trees of uniform intermediate (i.e. super-polynomial and sub-exponential) volume growth. We use this construction to provide first examples of unimodular random rooted trees of uniform intermediate growth, answering a question by Itai Benjamini. We find a peculiar change in structural properties for these trees at growth $r^{\log\log r}$.