Classify Tree Nodes in SQL

Write a PostgreSQL query that classifies every node in a tree as `Root`, `Inner`, or `Leaf`. ### Table `Tree` contains one row per node: | Column | Type | Meaning | | --- | --- | --- | | `id` | integer | Unique, non-null node identifier | | `pid` | integer, nullable | Parent node identifier, or `NULL` when the node has no parent | The input represents a nonempty tree. Each non-null `pid` refers to the `id` of its parent node. ### Classification rules - `Root`: the node has no parent, so its `pid` is `NULL`. - `Inner`: the node has a parent and at least one child. A child is a row whose `pid` equals this node's `id`. - `Leaf`: the node has a parent and no children. A root is classified as `Root` regardless of whether it has children, including when it is the tree's only node. ### Required output Return exactly one row per input node, with columns `id` and `type`. The `type` value must be exactly `Root`, `Inner`, or `Leaf` according to the rules above. Result rows may appear in any order. ### Example Input `Tree`: | id | pid | | --- | --- | | 1 | NULL | | 2 | 1 | | 3 | 1 | | 4 | 2 | Expected result, shown in identifier order only for readability: | id | type | | --- | --- | | 1 | Root | | 2 | Inner | | 3 | Leaf | | 4 | Leaf |