i think all you need is $t->close('/') before the mainloop

#!/usr/bin/perl
use strict;
use Tk;
use Tk::Tree;

my $mw = tkinit;


my $t = $mw->Scrolled(
    'Tree',
    -separator  => '/',
    -scrollbars => 'osoe',
)->pack;

foreach (qw(/ /a /b /c /a/s /a/s/d /a/s/d/f /b/n /b/n/m)) {
    my @text = reverse split( /\//, $_ );
    $t->add( $_, -text => @text[0] );
}


$t->autosetmode;
foreach ( $t->info( 'children', '/' ) ) {
    $t->close($_);
}

$mw->Button(-text=>'Close nodes',
            -command => sub{ closeTree($t, '/a') }
        )->pack();


$t->close('/');

MainLoop;


sub closeTree {
    my $tree = shift;
    my ($entryPath, $hideChildren) = @_;
    my @children = $tree->info (children => $entryPath);

    return if ! @children;

    for (@children) {
        closeTree ($tree, $_, 1);
        $tree->hide ('entry' => $_) if $hideChildren;
    }

    $tree->setmode ($entryPath, 'open') if length $entryPath;
}
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