Postgres recursive query with row_to_json, posz's attempt

  
json_tree

1
{
    "id": 1,
    "lvl": 0,
    "name": "name1",
    "children": [
        {
            "id": 3,
            "lvl": 1,
            "name": "name3",
            "children": [
                {
                    "id": 6,
                    "lvl": 2,
                    "name": "name6",
                    "children": [
                        {
                            "id": 8,
                            "lvl": 3,
                            "name": "name9",
                            "children": [
                            ],
                            "parent_id": 6,
                            "customer_id": 9,
                            "description": "1.3.6.9"
                        }
                    ],
                    "parent_id": 3,
                    "customer_id": 6,
                    "description": "1.3.6"
                },
                {
                    "id": 7,
                    "lvl": 2,
                    "name": "name7",
                    "children": [
                    ],
                    "parent_id": 3,
                    "customer_id": 7,
                    "description": "1.3.7"
                },
                {
                    "id": 9,
                    "lvl": 2,
                    "name": "name10",
                    "children": [
                    ],
                    "parent_id": 3,
                    "customer_id": 10,
                    "description": "1.3.10"
                }
            ],
            "parent_id": 1,
            "customer_id": 3,
            "description": "1.3"
        }
    ],
    "parent_id": null,
    "customer_id": 1,
    "description": "1"
}