Opencart：显示产品的所有子类别

I am working on the inbuilt google base extension in opencart 3.x, i need a special tag to display all the child element in a categories in format(Home>category1>category1_1)

Here's the inbuilt code i tried to tweak which displays every category one after the order

$categories = $this->model_catalog_product->getCategories($product['product_id']);

                    foreach ($categories as $category) {
                        $path = $this->getPath($category['category_id']);

                        if ($path) {
                            $string = '';

                            foreach (explode('_', $path) as $path_id) {
                                $category_info = $this->model_catalog_category->getCategory($path_id);

                                if ($category_info) {
                                    if (!$string) {
                                        $string = $category_info['name'];
                                    } else {
                                        $string .= ' > ' . $category_info['name'];
                                    }
                                }
                            }

                            $output .= '<g:product_type><![CDATA[' . $string . ']]></g:product_type>';
                        }
                    }