  The $Aldous\text{-}Broder$ and $Wilson$ are two well-known algorithms to
generate uniform spanning trees (USTs) based on random walks. This work studies
their relationship while they construct random trees with the goal of reducing
the total time required to build the spanning tree. Using the notion of
$branches$ $-$ paths generated by the two algorithms on particular stopping
times, we show that the trees built by the two algorithms when running on a
complete graph are statistically equivalent on these stopping times. This leads
to a hybrid algorithm that can generate uniform spanning trees of complete
graphs faster than either of the two algorithms. An efficient two-stage
framework is also proposed to explore this hybrid approach beyond complete
graphs, showing its feasibility in various examples, including transitive
graphs where it requires 25% less time than $Wilson$ to generate a UST.
