This is my solution. As I said above, I think it is less elegant than others published above, but I follow Roboticus's recommendation to publish it nonetheless.

#!/usr/bin/perl
use strict;
use warnings;

my $tree1 = [1, [2, [4, [7]], [5]], [3, [6, [8], [9]]]];
my $tree2 = [1, [2, [4, [7]], [5]], [3, [6, [8], [9]]]];
my $tree3 = [1, [2, [4, [7]], [5]], [3, [6, [9], [8]]]];

my $next_el1 = create_iterator($tree1);
my $next_el2 = create_iterator($tree3);

my $match = 1;


while (1) {
    my $left = $next_el1->();
    my $right = $next_el2->();
    no warnings 'uninitialized';
    print $left, " ", $right, "\n";
    unless ($left eq $right) {$match = 0 ; last} ;
    last unless defined $left;
}

if ($match) {
    print "The trees match \n";
} else {
    print "The trees don't match \n";
}

sub create_iterator {
    my $ref = shift;
    my @ref_list;
    return sub {
        while (ref $ref eq 'ARRAY') {
            push @ref_list, @$ref;
            $ref = shift @ref_list;
        } 
        my $leaf = $ref;
        $ref = shift @ref_list;
        return $leaf;
    }
}
[download]

Results comparing tree1 and tree3:

$ perl  fringe.pl
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 9
the trees don't match
[download]

And comparing tree1 and tree2:

$ perl  fringe.pl
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9

The trees match
[download]

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