How to mark checked the desired list item in a recursive function?

Question

PHP

Wrote a recursive function to display a list, with a choice of radio input.
The question arose of how to mark a certain element, so that when the list is displayed, it would be marked by default.

function tree($arr, $par = '0',){

    foreach ($arr as $key => $value){

      if ($value['parent']==$par){
          if ($key == '1') {
            $checked = 'checked';
          }

          $tree.= '<li><input type="radio" name="cat" value="'.$key.'" '.$checked.'>'.$value['name'];
        }
      
        $tree.= tree($arr,$key);	

        $tree.='</li>';
      }
    }

    return '<ul>'.$tree.'</ul>';
  }

When the condition that the key is equal to 1 is met, adds a checked to all top-level elements. And it turns out that the last element is marked, and not the one that is needed.
How to be?