How to create an AST parser that allows for syntax errors?

Question

Algorithms

For starters, how are AST (Abstract Syntax Tree) trees actually built? What to read?
How to create a parser for a language like SQL that generates an AST tree and forgives syntax errors?
Example: "3+4*5":

+
 / \
3   *
   / \
  4   5

But if the user made a mistake "3+4*+", the parser still tries to understand what the user meant:

+
 / \
3   *
   / \
  4   +
     / \
    ?   ?

Where to begin? Thank you!
SQL:

---SELECT_________________
   /           \           \
  .           FROM        JOIN
 / \           |         /    \
a city_name  people   address  ON
                                |
                                =______________
                               /               \
                              .____             .
                             /     \           / \
                            p  address_id     a  id

Answer 1

For starters, how are AST (Abstract Syntax Tree) trees actually built? What to read?

Dragon book.
Provide grammar rules for describing syntax errors and recovering from them. In bison, for example, there is an "error" token for this.