# MAKE SURE $str IS A _COPY_ OF YOUR DATA!!!
my $str = "/One/Two/Three";

my @paths = ();

while($str) {
    # If you don't want @push to include the current
    # cat (ie, "/One/Two/Three"), put this line after
    # the substr.  But then you'll have as the last
    # element an empty string.  Put a "last unless($str);"
    # between the substr and the push to avoid that.
    push @paths, $str;

    # Save everything from the begining of the string
    # up to, but not including, the last /.
    $str = substr($str, 0, rindex($str, '/'));
}

print "$_\n" for @paths;