Recursively flatten a multidimensional array into a 2d array

I'm trying to flat a multidimensional array to a given specific format. I have a tree that is saved as a nested array, which is ok, but the function given to render the array in the UI expects only one array and, per each child, an independent array. Instead of the nested array, each option should be at the same level.

This is how my var_dump of my array:

(
    [id] => 1
    [name] => some data.
    [emoji] => 🐕
    [parent_id] => 
    [children] => Array
        (
            [0] => Array
                (
                    [id] => 2
                    [name] => Food
                    [emoji] => 🥩
                    [parent_id] => 1
                    [children] => Array
                        (
                        )

                )

            [1] => Array
                (
                    [id] => 3
                    [name] => some other data
                    [emoji] => 😌
                    [parent_id] => 1
                    [children] => Array
                        (
                            [0] => Array
                                (
                                    [id] => 4
                                    [name] => Massages
                                    [emoji] => 💆
                                    [parent_id] => 3
                                    [children] => Array
                                        (
                                        )

                                )

                            [1] => Array
                                (
                                    [id] => 5
                                    [name] => Games
                                    [emoji] => 🎾
                                    [parent_id] => 3
                                    [children] => Array
                                        (
                                        )

                                )

                        )

                )
            )

        )
    )

And the expected result should be:

0] => Array
        (
            [id] => 1
            [name] => Rusty Corp.
            [emoji] => 🐕
            [parent_id] => 
        )

    [1] => Array
        (
            [id] => 2
            [name] => Food
            [emoji] => 🥩
            [parent_id] => 1
        )

    [2] => Array
        (
            [id] => 3
            [name] => Canine Therapy
            [emoji] => 😌
            [parent_id] => 1
        )

    [3] => Array
        (
            [id] => 4
            [name] => Massages
            [emoji] => 💆
            [parent_id] => 3
        )

    [4] => Array
        (
            [id] => 5
            [name] => Games
            [emoji] => 🎾
            [parent_id] => 3
        )

I tried different approaches like array_merge or custom flattening function but can't nail with the expected results, any suggestions?

Edit:

This is my flatten function:

private function flatten_array( array $array ) {
        $return = array();
        array_walk_recursive(
            $array,
            function( $a ) use ( &$return ) { 
                $return[] = $a;
            }
        );
        return $return;
    }

Here is a recursive function that will flatten the array, it will not take into account the null value of parent_id on the root element. Also the flattened array will start with the most nested elements at the start of the array and root element at the end.

function flatten_array($array, $flattened = []) {
    $current = [];
    foreach ($array as $key => $value) {
        if (is_array($value))
            $flattened = array_merge($flattened, flatten_array($value));
        else
            $current[$key] = $value;
    }
    $flattened[] = $current;
    return array_filter($flattened);
}