You can actually do this is a single statement. The technique is create a CTE to receives the parameters. In this case tree type and an array of owners. Then a second CTE to insert the tree_header and return the generated id. Finally, join the parameters with the with the resulting header. See Demo.
with parms (tree, owners) as
     ( values ('<tree type>', array ['<Owner_Array>']) )
   , make_tree as
     ( insert into tree_header(tree_type)
         select tree from parms
         returning tree_id, tree_type
     )
insert into  tree_detail(owner_name, tree_id)
   select unnest (p.Owner_Array), mt.tree_id
     from parms p 
     join make_tree mt on (mt.tree_type = p.tree);
Now since I do not write your obsurfication language (Python?) I am not exactly how it translates. But I guess something like:
import psycopg2
class MyData():
    def __init__(self, host="10.0.80.85", db="hhh", user="postgres", password="5bebfakee11"):
        self.conn = psycopg2.connect(host=host, database=db, user=user, password=password)
        self.cur = self.conn.cursor() 
            
    def close(self):
        self.cur.close()
        self.conn.close()
 db = MyData()
 statement = "with parms (tree, owners) as                   \
                  ( values ('%',  array [%]) )               \
                , make_tree as                               \
                  ( insert into tree_header(tree_type)       \
                      select tree from parms                 \ 
                      returning tree_id, tree_type           \ 
                  )                                          \
             insert into  tree_detail(owner_name, tree_id)   \
                select unnest (p.owners), mt.tree_id         \
                  from parms p                               \ 
                  join make_tree mt on (mt.tree_type = p.tree);"
 db.cur.execute(statement, (last_header_id,list_owner_array))
 
 db.conn.commit()
 db.close()
Note: There is nothing returned from the statement, it either works of Postgres throws and exception.