  For each $n \ge 1$, let $\mathrm{d}^n=(d^{n}(i),1 \le i \le n)$ be a sequence
of positive integers with even sum $\sum_{i=1}^n d^n(i) \ge 2n$. Let
$(G_n,T_n,\Gamma_n)$ be uniformly distributed over the set of simple graphs
$G_n$ with degree sequence $\mathrm{d}^n$, endowed with a spanning tree $T_n$
and rooted along an oriented edge $\Gamma_n$ of $G_n$ which is not an edge of
$T_n$. Under a finite variance assumption on degrees in $G_n$, we show that,
after rescaling, $T_n$ converges in distribution to the Brownian continuum
random tree as $n \to \infty$. Our main tool is a new version of Pitman's
additive coalescent (https://doi.org/10.1006/jcta.1998.2919), which can be used
to build both random trees with a fixed degree sequence, and random
tree-weighted graphs with a fixed degree sequence. As an input to the proof, we
also derive a Poisson approximation theorem for the number of loops and
multiple edges in the superposition of a fixed graph and a random graph with a
given degree sequence sampled according to the configuration model; we find
this to be of independent interest.
