List out subcategories under parentcategory based on product

I tried to display category that has subcategories within it. This is the sample structure of my categories:

Electronic
-Computer
-Phone
-Gadget

Grocery
-Food
-Drinks

This is my products table migration:

   Schema::create('products', function (Blueprint $table) {
        $table->increments('id');
        $table->string('name');
        $table->decimal('price')->nullable();
        $table->timestamps();
    });

This is my categories table migration

   Schema::create('categories', function (Blueprint $table) {
        $table->increments('id');
        $table->integer('parent_id')->nullable();
        $table->string('name')->nullable();
        $table->string('description')->nullable();
        $table->timestamps();
    });

and this one is the category_product which acts as for many to many table between category and product:

    Schema::create('category_product', function (Blueprint $table) {
        $table->increments('id');
        $table->integer('product_id')->unsigned()->index();
        $table->foreign('product_id')->references('id')->on('products')->onDelete('cascade');
        $table->integer('category_id')->unsigned()->index();
        $table->foreign('category_id')->references('id')->on('categories')->onDelete('cascade');
        $table->timestamps();
    });

I have setup all the relationship. Here are my models:

This is my category model:

class Category extends Model
{
    protected $table = 'categories';

    protected $fillable = [
        'name',
    ];
    public function products()
    {
        return $this->belongsToMany('App\Product');
    }

    public function parent()
    {
        return $this->belongsTo('App\Category', 'parent_id');
    }

    public function children()
    {
        return $this->hasMany('App\Category', 'parent_id');
    }

}

This is my product model:

class Product extends Model
{
   public function categories()
   {
     return $this->belongsToMany('App\Category');
   }
}

This is my ProductController.php, I'm able to display all parents category with its subcategories using this code:

public function show($id)
 {   
      $product = Product::findOrFail($id); 
      $categories = Category::with('children')->get();
      return view('products.show', compact('product','categories'));

 }

So my product.shows.blade looks like this:

@foreach($categories as $item)
    @if($item->children->count() > 0 )
        <li>
            {{ $item->name }}
            <ul>
                @foreach($item->children as $submenu)
                    <li>{{ $submenu->name }}</li>
                @endforeach
            </ul>
        </li>
    @endif

@endforeach

//OUTPUT:

Electronic  
 Computer
 Phone
 Gadget

Grocery
 Food
 Drinks

But let say that this particular product (called Product 1) has the parent category of Electronic and the subcategory is Computer and Phone which I already attached them in the database. This is the overview of the data from the database:

How do I display the category of Product 1 with its parent and subcategories? I want the output to be like

Product 1 

Category:
 Electronic
  Computer
  Phone

via Chebli Mohamed