  We study how to utilize (possibly erroneous) predictions in a model for
computing under uncertainty in which an algorithm can query unknown data. Our
aim is to minimize the number of queries needed to solve the minimum spanning
tree problem, a fundamental combinatorial optimization problem that has been
central also to the research area of explorable uncertainty. For all integral
$\gamma\ge 2$, we present algorithms that are $\gamma$-robust and
$(1+\frac{1}{\gamma})$-consistent, meaning that they use at most $\gamma OPT$
queries if the predictions are arbitrarily wrong and at most
$(1+\frac{1}{\gamma})OPT$ queries if the predictions are correct, where $OPT$
is the optimal number of queries for the given instance. Moreover, we show that
this trade-off is best possible. Furthermore, we argue that a suitably defined
hop distance is a useful measure for the amount of prediction error and design
algorithms with performance guarantees that degrade smoothly with the hop
distance. We also show that the predictions are PAC-learnable in our model. Our
results demonstrate that untrusted predictions can circumvent the known lower
bound of~$2$, without any degradation of the worst-case ratio. To obtain our
results, we provide new structural insights for the minimum spanning tree
problem that might be useful in the context of query-based algorithms
regardless of predictions. In particular, we generalize the concept of witness
sets -- the key to lower-bounding the optimum -- by proposing novel global
witness set structures and completely new ways of adaptively using those.
