  Petersen's theorem, one of the earliest results in graph theory, states that
any bridgeless cubic multigraph contains a perfect matching. While the original
proof was neither constructive nor algorithmic, Biedl, Bose, Demaine, and Lubiw
[J. Algorithms 38(1)] showed how to implement a later constructive proof by
Frink in $\mathcal{O}(n\log^{4}n)$ time using a fully dynamic
2-edge-connectivity structure. Then, Diks and Sta\'nczyk [SOFSEM 2010]
described a faster approach that only needs a fully dynamic connectivity
structure and works in $\mathcal{O}(n\log^{2}n)$ time. Both algorithms, while
reasonable simple, utilize non-trivial (2-edge-)connectivity structures. We
show that this is not necessary, and in fact a structure for maintaining a
dynamic tree, e.g. link-cut trees, suffices to obtain a simple
$\mathcal{O}(n\log n)$ time algorithm.
