A standard measure on trees on $n$ vertices is the Uniform Spanning Tree (UST) on the complete graph. This is the measure where every tree has equal probability, $1 / n^{n-2}$ by Cayley's formula.

Here is another measure. Take an Erdős–Rényi (i.e. edge independent) random graph $G \in G(n,p)$ with $p$ large enough to ensure that $G$ is asymptotically almost surely connected, and then choose a UST on $G$.

Note that if $p = 1$ these two measures are identical. My guess is that they are close (say in total variation distance), even for much smaller $p$. In particular, suppose that $$p \ge \frac{\log n + \omega}{n},$$ where $\omega \to \infty$ arbitrarily slowly as $n \to \infty$. (This is barely sufficient to guarantee that the probability that $G$ is connected tends to one.)

Are these actually the same measure in disguise? If not, can we say that they are "close" to the same measure, for example, by putting an upper bound on the total variation distance between the two measures that tends to zero as $n \to \infty$?