Recursively Conditionally get parent Id

I have two tables: Budget Line and Expense. They are structured in such a way that an expense must have either a parent record in the budget line table, or a parent record in the expense table. I need to select all child-most expenses for each budget line.

For example – BudgetLine:

Id	Description
1	TEST123
2	OTHERTEST

Expense:

Id	ParentId	ParentType	Description
1	1	BudgetLine	Group of Expenses
2	1	Expense	Expense # 1
3	1	Expense	Expense # 2
4	2	BudgetLine	Expense 3

Desired result:

BudgetLineId	ExpenseId	Description
1	2	Expense # 1
1	3	Expense # 2
2	4	Expense # 3

I am looking to omit expenses in the result only if they are the only sub-child. Note that an expense may have many children, grandchildren, etc.

I have tried the following, and researching various recursive CTE methods:

WITH RCTE AS
(
    SELECT Expense.Id, Expense.ParentId, Expense.ParentType, 1 AS Lvl, Expense.Id as startId FROM Expense 
    UNION ALL
    SELECT rh.Id, rh.ParentId, rh.ParentType, Lvl+1 AS Lvl, rc.Id as startId FROM dbo.Expense rh
    INNER JOIN RCTE rc ON rh.Id = rc.ParentId and rc.ParentType = 'Expense'
),
FilteredRCTE AS
(
    SELECT startId, MAX(LVL) AS Lvl 
    FROM RCTE 
    GROUP BY startID
),
RecursiveData AS
(
    SELECT FilteredRCTE.startId AS ExpenseId, RCTE.ParentId AS BudgetLineId 
    FROM FilteredRCTE 
    JOIN RCTE ON FilteredRCTE.startId = RCTE.startId AND FilteredRCTE.Lvl = RCTE.Lvl
)
SELECT * 
FROM RecursiveData

Which did in-fact obtain all the child Expenses and their associated parent BudgetLine, but it also included the middle-tier expenses (such as item 1 in the example) and I cannot figure out how to filter those middle-tier items out.

Here is a script to create tables / insert sample data:

CREATE TABLE [dbo].[BudgetLine]
(
    [Id] [int] IDENTITY(1,1) NOT NULL,
    [Description] [varchar](500) NULL,
) ON [PRIMARY]
GO

INSERT INTO dbo.BudgetLine VALUES ('TEST123')
INSERT INTO dbo.BudgetLine VALUES ('OTHERTEST')

CREATE TABLE [dbo].[Expense]
(
    [Id] [int] IDENTITY(1,1) NOT NULL,
    [ParentId] [int] NOT NULL,
    [ParentType] [varchar](100) NOT NULL,
    [Description] [varchar](max) NULL,
) ON [PRIMARY]
GO

INSERT INTO dbo.Expense VALUES ('1', 'BudgetLine', 'Group of Expenses')
INSERT INTO dbo.Expense VALUES ('1', 'Expense', 'Expense # 1')
INSERT INTO dbo.Expense VALUES ('1', 'Expense', 'Expense # 2')
INSERT INTO dbo.Expense VALUES ('2', 'BudgetLine', 'Expense # 3')

Answer

Maybe I have oversimplified, but the following returns your desired results, by checking that there is no other expense row connected to the current row.

WITH RCTE AS
(
    SELECT E.Id ExpenseId, E.ParentId, E.ParentType
    FROM #Expense E
    UNION ALL
    SELECT RH.Id, RH.ParentId, RH.ParentType
    FROM #Expense RH
    INNER JOIN RCTE RC ON RH.Id = RC.ParentId AND RC.ParentType = 'Expense'
)
SELECT *
FROM RCTE R1
WHERE NOT EXISTS (
    SELECT 1
    FROM RCTE R2
    WHERE R2.ParentId = R1.ExpenseId AND R2.ParentType = 'Expense'
);