A more Perlish cleaner version might look like:

use strict;
use warnings;

my @row = map {chomp; [split '']} <DATA>;
my $visible;

die "Data too awful to contemplate\n" if @row < 2 || @{$row[0]} < 2;

for my $col (1 .. $#{$row[0]} - 1) {
    my $tHigh = $row[0][$col];
    my $bHigh = $row[-1][$col];

    $visible += 2; # Edge trees always visible

    for my $depth (1 .. $#row - 1) {
        test(\$tHigh, \$row[$depth][$col]) if $tHigh != 9;
        test(\$bHigh, \$row[-$depth - 1][$col]) if $bHigh != 9;
        last if $tHigh == 9 && $bHigh == 9;
    }
}

for my $rowIdx (1 .. $#row - 1) {
    # Count down from top edge
    my $lHigh = $row[$rowIdx][0];
    my $rHigh = $row[$rowIdx][-1];

    $visible += 2; # Edge trees always visible

    for my $depth (1 .. $#row - 1) {
        test(\$lHigh, \$row[$rowIdx][$depth]) if $lHigh != 9;
        test(\$rHigh, \$row[$rowIdx][-$depth - 1]) if $rHigh != 9;
        last if $lHigh == 9 && $rHigh == 9;
    }
}

$visible += 4; # We didn't consider the corners so do that now.
print $visible;


sub test {
    my ($highest, $cell) = @_;

    return if $$highest >= $$cell;

    if ($$cell !~ /\./) {
        ++$visible;
        $$cell .= '.';
    }

    $$highest = $$cell;
    return $$highest == 9;
}


__DATA__
30373
25512
65332
33549
35390
[download]

which outputs 21 - the value suggested by the challenge. Note that is only deals correctly with rectangular forests containing 4 or more trees.