Fast algorithm to find one path through a tree

Fast algorithm to find one path through a tree

I have a finite rooted tree where each vertex has the same number of
children (except terminal vertices). Some edges are "free" and some are
"blocked". I am looking for an efficient algorithm to find just ONE path
from the root to any of the ends using "free" edges only, or determine
that none exist. Any ideas about the best way to do that? I wrote one
that's faster than checking all terminal nodes, but I am not sure if it's
the best.
I am more interested in the theoretical aspects of the different ways to
solve this problem, so a reference to a known algorithm or class of
algorithms would be most helpful.