What would an AST (abstract syntax tree) for an object-oriented programming language look like?

Question

I'm reading about AST (abstract syntax trees) but all the samples I see use expressions such as:

a + b * c

Which could be represented in a lispy like syntax as:

(+ a (* b c) )

Which will be the equivalent to:

  +
 / \
a   * 
   / \
  b   c

My question is How an AST for a class in a OOPL would look like?

My naive attempt is for this Java code:

 class Person { 
     String name;
     int    age;
     public String toString() { 
        return "name";
     }
 }

Is:

;Hand written
(classDeclaration Person 
     (varDeclaration String name)
     (varDeclaration int    age )
     (funcDeclaration String toString 
           (return "name")
     )
 )

But I'm not quite sure how close or far am I to a real AST representation.

Does it depends on the language I choose. How much detail is needed? Are those "xyzDeclaraction" needed or could be as:

 (Person (String name) (int age))

Where can I see a "real" representation of an actual programming language to learn more.

For example, see this: https://github.com/mono/mono/blob/master/mcs/mcs/class.cs — SK-logic, Jun 16 '11 at 18:41

aioobe · Accepted Answer · 2011-06-16T19:11:12.573

AST is an abstraction of the CST (concrete syntax tree, or, parse tree). The concrete syntax tree is the tree resulting from the productions (in the grammar) used to parse the file. So your AST is basically derived from your grammar definition, but has for transformed

                        Exp                    
                      /  |  \                   
                     /   |   \                       *
                 Ident BinOp Ident       into       / \
                  /      |     \                  "x" "y"
                 /       |      \
               "x"       *      "y"

All in all I think the example in your post looks fine. I would probably wrap the variable declarations in a varDeclList and the function declaration in a methDeclList, and the return statement in a stmtList. (See below.)

One more or less "real" representation of an AST is described by Apple in his book "Modern Compiler Implementation in Java". (Resources can be found here.)

Using those classes, your program would be represented as follows:

Program
    ClassDeclList
        ClassDecl
            Identifier
                id: Person
            VarDeclList
                VarDecl
                    type: String
                    id: name
                VarDecl
                    type: int
                    id: age
            MethDeclList
                MethodDecl
                    modifiers: public
                    returnType: String
                    id: toString
                    Formals
                        (empty)
                    StmtList
                        returnStmt
                            Identifier
                                id: name

I wonder if the `varDecl` and `methodDecl` nodes could be omitted since we are already in a `varDeclList` branch. Also how the inicialization for a final field would be represented? Would that be a new node `assignVal` or the value `varDecl`? — OscarRyz, Jun 20 '11 at 15:35
Hm.. As I see it `VarDeclList` etc are needed except in degenerate cases where there happens to be only one `VarDecl`. That is, a `ClassDecl` will (among other things) contain a `VarDeclList`. That's why I included it in the example tree. — aioobe, Jun 20 '11 at 17:14

score 13 · Answer 2 · edited May 23 '17 at 11:51

OP: Where can I see a real representation of an actual programming language to learn more?

For your source text as a file Person.java:

class Person {  
    String name;
    int    age;
    public String toString()
      { return "name";     } 
}

what follows are both Concrete and Abstract Syntax Tree in an S-expression-style dump of the parser tree from our DMS Software Reengineering Toolkit, using its Java1.6 parser. All the apparant complexity is pretty much caused by the real complexity of the language (e.g., of Java itself).

The CST clearly contains more stuff (139 nodes) than the AST (54 nodes). The AST drops everything that can be automatically inferred from the grammar, given the AST. This includes removing non-value-carrying leaves, unary productions, and compressing spines caused by left or right recursive grammar rules into explicit list nodes.

A left paren signals a new subtree. Following the left paren is the name of the node type; @Java~Java1_.6 might seem unnecessary until you understand DMS can handle many languages at once, including langauges nested inside one another. The #nnnnnn is the memory address of the node. ^M means "this node has M parents and is left off when M==1. Things inside [...] are the node value. A { M } means this list node has M list-children. Each node is stamped with position information.

This is the Concrete Syntax tree (see further down for AST):

(compilation_unit@Java~Java1_6=1#4885d00^0 Line 1 Column 1 File C:/temp/Person.java
 (type_declarations@Java~Java1_6=15#4885cc0 Line 1 Column 1 File C:/temp/Person.java
  (type_declarations@Java~Java1_6=16#4884d80 Line 1 Column 1 File C:/temp/Person.java)type_declarations
  (type_declaration@Java~Java1_6=17#4885ca0 Line 1 Column 1 File C:/temp/Person.java
   (type_class_modifiers@Java~Java1_6=77#4884dc0 Line 1 Column 1 File C:/temp/Person.java)type_class_modifiers
   (class_header@Java~Java1_6=89#4884ec0 Line 1 Column 1 File C:/temp/Person.java
   |('class'@Java~Java1_6=459#4884c60[Keyword:0] Line 1 Column 1 File C:/temp/Person.java)'class'
   |(IDENTIFIER@Java~Java1_6=447#4884e20[`Person'] Line 1 Column 7 File C:/temp/Person.java)IDENTIFIER
   |(type_parameters@Java~Java1_6=408#4884e80 Line 1 Column 14 File C:/temp/Person.java)type_parameters
   )class_header
   (class_body@Java~Java1_6=94#4885c80 Line 1 Column 14 File C:/temp/Person.java
   |('{'@Java~Java1_6=448#4884e60[Keyword:0] Line 1 Column 14 File C:/temp/Person.java)'{'
   |(class_body_declarations@Java~Java1_6=111#4885c60 Line 2 Column 5 File C:/temp/Person.java
   | (class_body_declarations@Java~Java1_6=111#4885380 Line 2 Column 5 File C:/temp/Person.java
   |  (class_body_declarations@Java~Java1_6=110#4885400 Line 2 Column 5 File C:/temp/Person.java
   |   (class_body_declaration@Java~Java1_6=118#4885360 Line 2 Column 5 File C:/temp/Person.java
   |   |(field_declaration@Java~Java1_6=168#4885440 Line 2 Column 5 File C:/temp/Person.java
   |   | (field_modifiers@Java~Java1_6=170#4884f40 Line 2 Column 5 File C:/temp/Person.java)field_modifiers
   |   | (type@Java~Java1_6=191#48852c0 Line 2 Column 5 File C:/temp/Person.java
   |   |  (name@Java~Java1_6=406#48851e0 Line 2 Column 5 File C:/temp/Person.java
   |   |   (IDENTIFIER@Java~Java1_6=447#4884f20[`String'] Line 2 Column 5 File C:/temp/Person.java)IDENTIFIER
   |   |   (type_arguments@Java~Java1_6=407#4885160 Line 2 Column 12 File C:/temp/Person.java)type_arguments
   |   |  )name
   |   |  (brackets@Java~Java1_6=157#4885260 Line 2 Column 12 File C:/temp/Person.java)brackets
   |   | )type
   |   | (variable_declarator_list@Java~Java1_6=179#4884e00 Line 2 Column 12 File C:/temp/Person.java
   |   |  (variable_declarator@Java~Java1_6=181#4885300 Line 2 Column 12 File C:/temp/Person.java
   |   |   (variable_declarator_id@Java~Java1_6=167#4885320 Line 2 Column 12 File C:/temp/Person.java
   |   |   |(IDENTIFIER@Java~Java1_6=447#4885140[`name'] Line 2 Column 12 File C:/temp/Person.java)IDENTIFIER
   |   |   |(brackets@Java~Java1_6=157#4885040 Line 2 Column 16 File C:/temp/Person.java)brackets
   |   |   )variable_declarator_id
   |   |  )variable_declarator
   |   | )variable_declarator_list
   |   | (';'@Java~Java1_6=440#4885100[Keyword:0] Line 2 Column 16 File C:/temp/Person.java)';'
   |   |)field_declaration
   |   )class_body_declaration
   |  )class_body_declarations
   |  (class_body_declaration@Java~Java1_6=118#48852e0 Line 3 Column 5 File C:/temp/Person.java
   |   (field_declaration@Java~Java1_6=168#4885480 Line 3 Column 5 File C:/temp/Person.java
   |   |(field_modifiers@Java~Java1_6=170#4885340 Line 3 Column 5 File C:/temp/Person.java)field_modifiers
   |   |(type@Java~Java1_6=192#4885220 Line 3 Column 5 File C:/temp/Person.java
   |   | (primitive_type@Java~Java1_6=198#4885420 Line 3 Column 5 File C:/temp/Person.java
   |   |  ('int'@Java~Java1_6=479#48853e0[Keyword:0] Line 3 Column 5 File C:/temp/Person.java)'int'
   |   | )primitive_type
   |   | (brackets@Java~Java1_6=157#4885200 Line 3 Column 12 File C:/temp/Person.java)brackets
   |   |)type
   |   |(variable_declarator_list@Java~Java1_6=179#4885540 Line 3 Column 12 File C:/temp/Person.java
   |   | (variable_declarator@Java~Java1_6=181#4885520 Line 3 Column 12 File C:/temp/Person.java
   |   |  (variable_declarator_id@Java~Java1_6=167#4885500 Line 3 Column 12 File C:/temp/Person.java
   |   |   (IDENTIFIER@Java~Java1_6=447#4884fc0[`age'] Line 3 Column 12 File C:/temp/Person.java)IDENTIFIER
   |   |   (brackets@Java~Java1_6=157#48854e0 Line 3 Column 15 File C:/temp/Person.java)brackets
   |   |  )variable_declarator_id
   |   | )variable_declarator
   |   |)variable_declarator_list
   |   |(';'@Java~Java1_6=440#48854c0[Keyword:0] Line 3 Column 15 File C:/temp/Person.java)';'
   |   )field_declaration
   |  )class_body_declaration
   | )class_body_declarations
   | (class_body_declaration@Java~Java1_6=117#4885c40 Line 4 Column 5 File C:/temp/Person.java
   |  (method_declaration@Java~Java1_6=135#4885c00 Line 4 Column 5 File C:/temp/Person.java
   |   (method_modifiers@Java~Java1_6=141#4885700 Line 4 Column 5 File C:/temp/Person.java
   |   |(method_modifiers@Java~Java1_6=142#4884e40 Line 4 Column 5 File C:/temp/Person.java)method_modifiers
   |   |(method_modifier@Java~Java1_6=147#48856a0 Line 4 Column 5 File C:/temp/Person.java
   |   | ('public'@Java~Java1_6=453#48853a0[Keyword:0] Line 4 Column 5 File C:/temp/Person.java)'public'
   |   |)method_modifier
   |   )method_modifiers
   |   (type_parameters@Java~Java1_6=408#4885740 Line 4 Column 12 File C:/temp/Person.java)type_parameters
   |   (type@Java~Java1_6=191#4885900 Line 4 Column 12 File C:/temp/Person.java
   |   |(name@Java~Java1_6=406#48852a0 Line 4 Column 12 File C:/temp/Person.java
   |   | (IDENTIFIER@Java~Java1_6=447#4885660[`String'] Line 4 Column 12 File C:/temp/Person.java)IDENTIFIER
   |   | (type_arguments@Java~Java1_6=407#48851a0 Line 4 Column 19 File C:/temp/Person.java)type_arguments
   |   |)name
   |   |(brackets@Java~Java1_6=157#48858c0 Line 4 Column 19 File C:/temp/Person.java)brackets
   |   )type
   |   (IDENTIFIER@Java~Java1_6=447#48855c0[`toString'] Line 4 Column 19 File C:/temp/Person.java)IDENTIFIER
   |   (parameters@Java~Java1_6=158#48858e0 Line 4 Column 27 File C:/temp/Person.java
   |   |('('@Java~Java1_6=450#4885840[Keyword:0] Line 4 Column 27 File C:/temp/Person.java)'('
   |   |(')'@Java~Java1_6=451#4885620[Keyword:0] Line 4 Column 28 File C:/temp/Person.java)')'
   |   )parameters
   |   (brackets@Java~Java1_6=157#4885060 Line 5 Column 7 File C:/temp/Person.java)brackets
   |   (block@Java~Java1_6=217#4885be0 Line 5 Column 7 File C:/temp/Person.java
   |   |('{'@Java~Java1_6=448#48851c0[Keyword:0] Line 5 Column 7 File C:/temp/Person.java)'{'
   |   |(statement_sequence@Java~Java1_6=218#4885ba0 Line 5 Column 9 File C:/temp/Person.java
   |   | (statement_sequence_member@Java~Java1_6=223#4885b80 Line 5 Column 9 File C:/temp/Person.java
   |   |  (executable_statement@Java~Java1_6=243#4885b60 Line 5 Column 9 File C:/temp/Person.java
   |   |   ('return'@Java~Java1_6=491#4884f60[Keyword:0] Line 5 Column 9 File C:/temp/Person.java)'return'
   |   |   (expression@Java~Java1_6=332#4885ac0 Line 5 Column 16 File C:/temp/Person.java
   |   |   |(conditional_expression@Java~Java1_6=345#4885a60 Line 5 Column 16 File C:/temp/Person.java
   |   |   | (conditional_or_expression@Java~Java1_6=347#4885a20 Line 5 Column 16 File C:/temp/Person.java
   |   |   |  (conditional_and_expression@Java~Java1_6=349#48859e0 Line 5 Column 16 File C:/temp/Person.java
   |   |   |   (inclusive_or_expression@Java~Java1_6=351#48857e0 Line 5 Column 16 File C:/temp/Person.java
   |   |   |   |(exclusive_or_expression@Java~Java1_6=353#48855a0 Line 5 Column 16 File C:/temp/Person.java
   |   |   |   | (and_expression@Java~Java1_6=355#4885940 Line 5 Column 16 File C:/temp/Person.java
   |   |   |   |  (equality_expression@Java~Java1_6=357#4885880 Line 5 Column 16 File C:/temp/Person.java
   |   |   |   |   (relational_expression@Java~Java1_6=360#4885800 Line 5 Column 16 File C:/temp/Person.java
   |   |   |   |   |(shift_expression@Java~Java1_6=366#48856c0 Line 5 Column 16 File C:/temp/Person.java
   |   |   |   |   | (additive_expression@Java~Java1_6=370#4885180 Line 5 Column 16 File C:/temp/Person.java
   |   |   |   |   |  (multiplicative_expression@Java~Java1_6=373#4885780 Line 5 Column 16 File C:/temp/Person.java
   |   |   |   |   |   (unary_expression@Java~Java1_6=383#4885600 Line 5 Column 16 File C:/temp/Person.java
   |   |   |   |   |   |(unary_expression_not_plus_minus@Java~Java1_6=389#4885680 Line 5 Column 16 File C:/temp/Person.java
   |   |   |   |   |   | (literal@Java~Java1_6=390#4884f80 Line 5 Column 16 File C:/temp/Person.java
   |   |   |   |   |   |  (STRING@Java~Java1_6=536#4885120[`name'] Line 5 Column 16 File C:/temp/Person.java)STRING
   |   |   |   |   |   | )literal
   |   |   |   |   |   |)unary_expression_not_plus_minus
   |   |   |   |   |   )unary_expression
   |   |   |   |   |  )multiplicative_expression
   |   |   |   |   | )additive_expression
   |   |   |   |   |)shift_expression
   |   |   |   |   )relational_expression
   |   |   |   |  )equality_expression
   |   |   |   | )and_expression
   |   |   |   |)exclusive_or_expression
   |   |   |   )inclusive_or_expression
   |   |   |  )conditional_and_expression
   |   |   | )conditional_or_expression
   |   |   |)conditional_expression
   |   |   )expression
   |   |   (';'@Java~Java1_6=440#48856e0[Keyword:0] Line 5 Column 22 File C:/temp/Person.java)';'
   |   |  )executable_statement
   |   | )statement_sequence_member
   |   |)statement_sequence
   |   |('}'@Java~Java1_6=449#4885b40[Keyword:0] Line 5 Column 28 File C:/temp/Person.java)'}'
   |   )block
   |  )method_declaration
   | )class_body_declaration
   |)class_body_declarations
   |('}'@Java~Java1_6=449#4885bc0[Keyword:0] Line 6 Column 1 File C:/temp/Person.java)'}'
   )class_body
  )type_declaration
 )type_declarations
 (optional_CONTROL_Z@Java~Java1_6=5#4885ce0 Line 7 Column 1 File C:/temp/Person.java)optional_CONTROL_Z
)compilation_unit

This is the AST (automatically generated by DMS from the CST):

(compilation_unit@Java~Java1_6=1#486f900^0 Line 1 Column 1 File C:/temp/Person.java
 (type_declarations@Java~Java1_6=15#486f4c0 {1} Line 1 Column 1 File C:/temp/Person.java
  (type_declaration@Java~Java1_6=17#486f5e0 Line 1 Column 1 File C:/temp/Person.java
   (type_class_modifiers@Java~Java1_6=77#486eda0 Line 1 Column 1 File C:/temp/Person.java)type_class_modifiers
   (class_header@Java~Java1_6=89#486ee60 Line 1 Column 1 File C:/temp/Person.java
   |(IDENTIFIER@Java~Java1_6=447#486ede0[`Person'] Line 1 Column 7 File C:/temp/Person.java)IDENTIFIER
   |(type_parameters@Java~Java1_6=408#486ee20 Line 1 Column 14 File C:/temp/Person.java)type_parameters
   )class_header
   (class_body@Java~Java1_6=94#486f040 Line 1 Column 14 File C:/temp/Person.java
   |(class_body_declarations@Java~Java1_6=111#486ee40 {3} Line 2 Column 5 File C:/temp/Person.java
   | (class_body_declaration@Java~Java1_6=118#486f300 Line 2 Column 5 File C:/temp/Person.java
   |  (field_declaration@Java~Java1_6=168#486f380 Line 2 Column 5 File C:/temp/Person.java
   |   (field_modifiers@Java~Java1_6=170#486eec0 Line 2 Column 5 File C:/temp/Person.java)field_modifiers
   |   (type@Java~Java1_6=191#486f240 Line 2 Column 5 File C:/temp/Person.java
   |   |(name@Java~Java1_6=406#486f180 Line 2 Column 5 File C:/temp/Person.java
   |   | (IDENTIFIER@Java~Java1_6=447#486eea0[`String'] Line 2 Column 5 File C:/temp/Person.java)IDENTIFIER
   |   | (type_arguments@Java~Java1_6=407#486f0e0 Line 2 Column 12 File C:/temp/Person.java)type_arguments
   |   |)name
   |   |(brackets@Java~Java1_6=157#486f200 Line 2 Column 12 File C:/temp/Person.java)brackets
   |   )type
   |   (variable_declarator@Java~Java1_6=181#486ef20 Line 2 Column 12 File C:/temp/Person.java
   |   |(variable_declarator_id@Java~Java1_6=167#486efe0 Line 2 Column 12 File C:/temp/Person.java
   |   | (IDENTIFIER@Java~Java1_6=447#486f0c0[`name'] Line 2 Column 12 File C:/temp/Person.java)IDENTIFIER
   |   | (brackets@Java~Java1_6=157#486f060 Line 2 Column 16 File C:/temp/Person.java)brackets
   |   |)variable_declarator_id
   |   )variable_declarator
   |  )field_declaration
   | )class_body_declaration
   | (class_body_declaration@Java~Java1_6=118#486f000 Line 3 Column 5 File C:/temp/Person.java
   |  (field_declaration@Java~Java1_6=168#486f320 Line 3 Column 5 File C:/temp/Person.java
   |   (field_modifiers@Java~Java1_6=170#486f2a0 Line 3 Column 5 File C:/temp/Person.java)field_modifiers
   |   (type@Java~Java1_6=192#486eee0 Line 3 Column 5 File C:/temp/Person.java
   |   |(primitive_type@Java~Java1_6=198#486ef60 Line 3 Column 5 File C:/temp/Person.java)primitive_type
   |   |(brackets@Java~Java1_6=157#486ee00 Line 3 Column 12 File C:/temp/Person.java)brackets
   |   )type
   |   (variable_declarator@Java~Java1_6=181#486f2c0 Line 3 Column 12 File C:/temp/Person.java
   |   |(variable_declarator_id@Java~Java1_6=167#486f3a0 Line 3 Column 12 File C:/temp/Person.java
   |   | (IDENTIFIER@Java~Java1_6=447#486f120[`age'] Line 3 Column 12 File C:/temp/Person.java)IDENTIFIER
   |   | (brackets@Java~Java1_6=157#486ef00 Line 3 Column 15 File C:/temp/Person.java)brackets
   |   |)variable_declarator_id
   |   )variable_declarator
   |  )field_declaration
   | )class_body_declaration
   | (class_body_declaration@Java~Java1_6=117#486f7a0 Line 4 Column 5 File C:/temp/Person.java
   |  (method_declaration@Java~Java1_6=135#486f480 Line 4 Column 5 File C:/temp/Person.java
   |   (method_modifiers@Java~Java1_6=141#486f460 {1} Line 4 Column 5 File C:/temp/Person.java
   |   |(method_modifier@Java~Java1_6=147#486f400 Line 4 Column 5 File C:/temp/Person.java)method_modifier
   |   )method_modifiers
   |   (type_parameters@Java~Java1_6=408#486f540 Line 4 Column 12 File C:/temp/Person.java)type_parameters
   |   (type@Java~Java1_6=191#486f740 Line 4 Column 12 File C:/temp/Person.java
   |   |(name@Java~Java1_6=406#486f620 Line 4 Column 12 File C:/temp/Person.java
   |   | (IDENTIFIER@Java~Java1_6=447#486f080[`String'] Line 4 Column 12 File C:/temp/Person.java)IDENTIFIER
   |   | (type_arguments@Java~Java1_6=407#486f640 Line 4 Column 19 File C:/temp/Person.java)type_arguments
   |   |)name
   |   |(brackets@Java~Java1_6=157#486f700 Line 4 Column 19 File C:/temp/Person.java)brackets
   |   )type
   |   (IDENTIFIER@Java~Java1_6=447#486f140[`toString'] Line 4 Column 19 File C:/temp/Person.java)IDENTIFIER
   |   (parameters@Java~Java1_6=158#486f760 Line 4 Column 27 File C:/temp/Person.java)parameters
   |   (brackets@Java~Java1_6=157#486f820 Line 5 Column 7 File C:/temp/Person.java)brackets
   |   (block@Java~Java1_6=217#486f780 Line 5 Column 7 File C:/temp/Person.java
   |   |(statement_sequence@Java~Java1_6=218#486f6e0 Line 5 Column 9 File C:/temp/Person.java
   |   | (statement_sequence_member@Java~Java1_6=223#486f6c0 Line 5 Column 9 File C:/temp/Person.java
   |   |  (executable_statement@Java~Java1_6=243#486f6a0 Line 5 Column 9 File C:/temp/Person.java
   |   |   (unary_expression_not_plus_minus@Java~Java1_6=389#486f720 Line 5 Column 16 File C:/temp/Person.java
   |   |   |(literal@Java~Java1_6=390#486f280 Line 5 Column 16 File C:/temp/Person.java
   |   |   | (STRING@Java~Java1_6=536#486f160[`name'] Line 5 Column 16 File C:/temp/Person.java)STRING
   |   |   |)literal
   |   |   )unary_expression_not_plus_minus
   |   |  )executable_statement
   |   | )statement_sequence_member
   |   |)statement_sequence
   |   )block
   |  )method_declaration
   | )class_body_declaration
   |)class_body_declarations
   )class_body
  )type_declaration
 )type_declarations
 (optional_CONTROL_Z@Java~Java1_6=5#486f4e0 Line 7 Column 1 File C:/temp/Person.java)optional_CONTROL_Z
)compilation_unit

EDIT March 2015: Here's a link to some C++ AST examples

Edit May 2015: DMS has long done Java 1.7 and 1.8, too.

Exactly which DMS product did you use to generate the Java AST? I'm trying to download it, and I see products **Test (Code) Coverage**, **Source Code Browser**, **Formatter / Obfuscator**, etc., but no generic "parser." — thatWiseGuy, Dec 28 '15 at 22:32
@thatWiseGuy Almost all SD products related to Java will construct that AST as they go about their business by virtue of containing the DMS parser for Java. Those products won't *give* you that AST; they are focused on the task they are designed to do (e.g., collect test coverage, format/obfuscate, etc.). The one product that will give you an AST as a manipulable data stucture, and/or as XML output, is the DMS Software Reengineering Toolkit ("DMS") www.semanticdesigns.com/Products/DMS/DMSToolkit.html with its Java front end. DMS does not have a trial download, contact SD for more details — Ira Baxter, Dec 28 '15 at 22:58

score 2 · Answer 3 · answered Jun 16 '11 at 18:25

2

Take a look at the Eclipse JDT AST implementation.

As first introduction you could read this tutorial, too.

answered Jun 16 '11 at 18:25

Omnaest

3,103
1
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link to tutorial is broken, a fix would be nice please – Lorenzo Apr 05 '21 at 11:53

What would an AST (abstract syntax tree) for an object-oriented programming language look like?

3 Answers3

Linked