My first question here, please correct if necessary.

Simply enumerate unique unary-binary trees as related to Motzkin numbers ([OEIS][1]) ([Wikipedia][2]).
 
Examples:

    M(1)
    t
    
    M(2)
    (u t)
    
    M(3)
    (u (u t))
    (b t t)
    
    M(4)
    (u (u (u t)))
    (u (b t t))
    (b t (u t))
    (b (u t) t)
    
    M(5)
    (u (u (u (u t))))
    (u (u (b t t)))
    (u (b t (u t)))
    (u (b (u t) t))
    (b t (u (u t)))
    (b t (b t t))
    (b (u t) (u t))
    (b (u (u t)) t)
    (b (b t t) t)

These examples use prefix notation, where `b`inary trees have 2 branches succeeding them (in the last example, `(b t t)` and `t`), and `u`nary branches have only 1 (in the second to last example,`(u t)` follows the first `u`).

A possible place to check for duplicate trees
-

One place to check for a duplicate is with M(5) which if you do it wrong will give you 

    (b (u t) (u t))

twice.

This one tree was generated twice for M(5) from M(4) trees  

    (b (u t) (u t))  

the first by adding a unary branch to

    (b (u t) t)

and second by adding a unary branch to

    (b t (u t))

Syntax
-

The syntax does not have to be exactly the same as the examples; if others understand that they are the same and can translate from one version to another that is acceptable. e.g. using `[]` instead of `()`, extra level of brackets `[[]]` instead of `[]`, outer parenthesis are present or missing, extra commas or no commas, extra spaces, etc.


  [1]: https://oeis.org/A001006
  [2]: https://en.wikipedia.org/wiki/Motzkin_number