def union_by_name(df1: DataFrame, df2: DataFrame, allow_missing_columns: bool) -> DataFrame:
    l_cols = set(df1.column_names)
    r_cols = set(df2.column_names)
    on = [col(c) for c in l_cols.intersection(r_cols)]
    how = "outer" if allow_missing_columns else "inner"
    return df1.join(df2, left_on=on, right_on=on, how=how)

import daft

df1 = daft.from_pydict({"a": ["x", "y", "S"], "b": [1, 2, 3]})
df2 = daft.from_pydict({"a": ["x", "y", "T"]})

union_by_name(df1, df2, allow_missing_columns=True).show()
"""
╭──────┬───────╮
│ a    ┆ b     │
│ ---  ┆ ---   │
│ Utf8 ┆ Int64 │
╞══════╪═══════╡
│ x    ┆ 1     │
├╌╌╌╌╌╌┼╌╌╌╌╌╌╌┤
│ y    ┆ 2     │
├╌╌╌╌╌╌┼╌╌╌╌╌╌╌┤
│ S    ┆ 3     │
├╌╌╌╌╌╌┼╌╌╌╌╌╌╌┤
│ T    ┆ None  │
╰──────┴───────╯

(Showing first 4 of 4 rows)
"""

def _concat_by_name(df1: daft.DataFrame, df2: daft.DataFrame) -> daft.DataFrame:
    df1_cols = set((x.name, x.dtype) for x in df1.schema())
    df2_cols = set((x.name, x.dtype) for x in df2.schema())
    all_cols = [name for name, _ in df1_cols.union(df2_cols)]

    return (
        df1.with_columns(
            {name: daft.lit(None).cast(dt) for name, dt in df2_cols - df1_cols}
        ).select(*all_cols)
    ).concat(
        df2.with_columns(
            {name: daft.lit(None).cast(dt) for name, dt in df1_cols - df2_cols}
        ).select(*all_cols)
    )