I don't know if you can disable the warning without modifying the source code, but one way to capture the warning is to use eval

use strict;
use warnings;
use Tree;

my ($i, $T, $n);
$T = Tree->new('root');
my $t = $T;
for($i=0;$i<100;$i++){
        $n = Tree->new("child $i");
        eval { $t->add_child({}, $n); };
        if ($@) {
            # take some action...
        }
        $t = $n;
}
print "success\n";
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Then you can decide what action to take.

My guess is that the NONFATAL attempt did not work because it is in the wrong scope; NONFATAL would have to be used inside the Tree module.