Hmm, I've not used it myself, but the docs seemed pretty clear that the file/filehandle is for reading, and that Bio::TreeIO doesn't have a print method, but relies on you to print the contents of the tree. From the DESCRIPTION:

This is the driver module for Tree reading from data streams and flatf
+iles. This is intended to be able to create Bio::Tree::TreeI objects.
[download]

and the SYNOPSIS:

{
      use Bio::TreeIO;
      my $treeio = Bio::TreeIO->new(-format => 'newick',
                                    -file   => 'globin.dnd');
      while( my $tree = $treeio->next_tree ) {
                print "Tree is ", $tree->number_nodes, "\n";
      }
  }
[download]

In this case the OP has created his tree with Bio::TreeIO and it's held in $treeout.