This is how you can use recursive relations:

public function childrenAccounts()
{
    return $this->hasMany('Account', 'act_parent', 'act_id');
}

public function allChildrenAccounts()
{
    return $this->childrenAccounts()->with('allChildrenAccounts');
}

Then:

$account = Account::with('allChildrenAccounts')->first();

$account->allChildrenAccounts; // collection of recursively loaded children
// each of them having the same collection of children:
$account->allChildrenAccounts->first()->allChildrenAccounts; // .. and so on

This way you save a lot of queries. This will execute 1 query per each nesting level + 1 additional query.

I can't guarantee it will be efficient for your data, you need to test it definitely.

---

This is for childless accounts:

public function scopeChildless($q)
{
   $q->has('childrenAccounts', '=', 0);
}

then:

$childlessAccounts = Account::childless()->get();