# Expression trees

When parsing an expression via `math.parse(expr)`, math.js generates an
expression tree and returns the root node of the tree. An expression tree can
be used to analyze, manipulate, and evaluate expressions.

Example:

```js
const node = math.parse('sqrt(2 + x)')
```

In this case, the expression `sqrt(2 + x)` is parsed as:

```
  FunctionNode    sqrt
                   |
  OperatorNode     +
                  / \
  ConstantNode   2   x   SymbolNode
```

Alternatively, this expression tree can be built by manually creating nodes:

```js
const node1 = new math.ConstantNode(2)
const node2 = new math.SymbolNode('x')
const node3 = new math.OperatorNode('+', 'add', [node1, node2])
const node4 = new math.FunctionNode('sqrt', [node3])
```

The resulting expression tree with root node `node4` is equal to the expression
tree generated by `math.parse('sqrt(2 + x)')`.


## API

### Methods

All nodes have the following methods:

-   `clone() : Node`

    Create a shallow clone of the node.
    The node itself is cloned, its childs are not cloned.

-   `cloneDeep() : Node`

    Create a deep clone of the node.
    Both the node as well as all its childs are cloned recursively.

-   `compile() : Object`

    Compile an expression into optimized JavaScript code. `compile` returns an
    object with a function `evaluate([scope])` to evaluate. Example:

    ```js
    const node = math.parse('2 + x')   // returns the root Node of an expression tree
    const code = node.compile()        // returns {evaluate: function (scope) {...}}
    const evaluate = code.evaluate({x: 3}) // returns 5
    ```

-   `evaluate([scope]) : Object`

    Compile and evaluate an expression, this is the equivalent of doing
    `node.compile().evaluate(scope)`. Example:

    ```js
    const node = math.parse('2 + x')   // returns the root Node of an expression tree
    const evaluate = node.evaluate({x: 3}) // returns 5
    ```

-   `equals(other: Node) : boolean`

    Test whether this node equals an other node. Does a deep comparison of the
    values of both nodes.

-   `filter(callback: function) : Node[]`

    Recursively filter nodes in an expression tree. The `callback` function is
    called as `callback(node: Node, path: string, parent: Node) : boolean` for
    every node in the tree, and must return a boolean. The function `filter`
    returns an array with nodes for which the test returned true.
    Parameter `path` is a string containing a relative JSON Path.

    Example:

    ```js
    const node = math.parse('x^2 + x/4 + 3*y')
    const filtered = node.filter(function (node) {
      return node.isSymbolNode && node.name === 'x'
    })
    // returns an array with two entries: two SymbolNodes 'x'
    ```

-   `forEach(callback: function) : void`

    Execute a callback for each of the child nodes of this node. The `callback`
    function is called as `callback(child: Node, path: string, parent: Node): void`.
    Parameter `path` is a string containing a relative JSON Path.

    See also `traverse`, which is a recursive version of `forEach`.

    Example:

    ```js
    const node = math.parse('3 * x + 2')
    node.forEach(function (node, path, parent) {
      switch (node.type) {
        case 'OperatorNode':
          console.log(node.type, node.op)
          break
        case 'ConstantNode':
          console.log(node.type, node.value)
          break
        case 'SymbolNode':
          console.log(node.type, node.name)
          break
        default:
          console.log(node.type)
      }
    })
    // outputs:
    //   OperatorNode *
    //   ConstantNode 2
    ```

-   `map(callback: function) : Node[]`

    Transform a node. Creates a new Node having it's childs be the results of
    calling the provided callback function for each of the childs of the original
    node. The `callback` function is called as `callback(child: Node, path: string,
    parent: Node)` and must return a Node. Parameter `path` is a string containing
    a relative JSON Path.

    See also `transform`, which is a recursive version of `map`.

-	`toHTML(options: object): string`

	Get a HTML representation of the parsed expression. Example:

    ```js
    const node = math.parse('sqrt(2/3)')
    node.toHTML()
	// returns
	// <span class="math-function">sqrt</span>
	// <span class="math-paranthesis math-round-parenthesis">(</span>
	// <span class="math-number">2</span>
	// <span class="math-operator math-binary-operator math-explicit-binary-operator">/</span>
	// <span class="math-number">3</span>
	// <span class="math-paranthesis math-round-parenthesis">)</span>
    ```

    Information about the available HTML classes in [HTML Classes](html_classes.md).
    Information about the options in [Customization](customization.md#custom-html-latex-and-string-output).

-   `toString(options: object) : string`

    Get a string representation of the parsed expression. This is not exactly
    the same as the original input. Example:

    ```js
    const node = math.parse('3+4*2')
    node.toString()  // returns '3 + (4 * 2)'
    ```

    Information about the options in [Customization](customization.md#custom-html-latex-and-string-output).

-   `toTex(options: object): string`

    Get a [LaTeX](https://en.wikipedia.org/wiki/LaTeX) representation of the
    expression. Example:

    ```js
    const node = math.parse('sqrt(2/3)')
    node.toTex() // returns '\sqrt{\frac{2}{3}}'
    ```

    Information about the options in [Customization](customization.md#custom-html-latex-and-string-output).

-   `transform(callback: function)`

    Recursively transform an expression tree via a transform function. Similar
    to `Array.map`, but recursively executed on all nodes in the expression tree.
    The callback function is a mapping function accepting a node, and returning
    a replacement for the node or the original node. Function `callback` is
    called as `callback(node: Node, path: string, parent: Node)` for every node
    in the tree, and must return a `Node`. Parameter `path` is a string containing
    a relative JSON Path.

    The transform function will stop iterating when a node is replaced by the
    callback function, it will not iterate over replaced nodes.

    For example, to replace all nodes of type `SymbolNode` having name 'x' with a
    ConstantNode with value `3`:

    ```js
    const node = math.parse('x^2 + 5*x')
    const transformed = node.transform(function (node, path, parent) {
      if (node.isSymbolNode && node.name === 'x') {
        return new math.ConstantNode(3)
      }
      else {
        return node
      }
    })
    transformed.toString() // returns '3 ^ 2 + 5 * 3'
    ```

-   `traverse(callback: function): void`

    Recursively traverse all nodes in a node tree. Executes given callback for
    this node and each of its child nodes. Similar to `Array.forEach`, except
    recursive.
    The callback function is a mapping function accepting a node, and returning
    nothing. Function `callback` is
    called as `callback(node: Node, path: string, parent: Node)` for every node
    in the tree. Parameter `path` is a string containing a relative JSON Path.
    Example:

    ```js
    const node = math.parse('3 * x + 2')
    node.traverse(function (node, path, parent) {
      switch (node.type) {
        case 'OperatorNode':
          console.log(node.type, node.op)
          break
        case 'ConstantNode':
          console.log(node.type, node.value)
          break
        case 'SymbolNode':
          console.log(node.type, node.name)
          break
        default:
          console.log(node.type)
      }
    })
    // outputs:
    //   OperatorNode +
    //   OperatorNode *
    //   ConstantNode 3
    //   SymbolNode x
    //   ConstantNode 2
    ```


### Properties

Each `Node` has the following properties:

-   `comment: string`

    A string holding a comment if there was any in the expression, or else the
    string will be empty string. A comment can be attached to the root node of
    an expression or to each of the childs nodes of a `BlockNode`.

-   `isNode: true`

    Is defined with value `true` on Nodes. Additionally, each type of node
    adds it's own flag, for example a `SymbolNode` as has a property
    `isSymbolNode: true`.

-   `type: string`

    The type of the node, for example `'SymbolNode'` in case of a `SymbolNode`.


## Nodes

math.js has the following types of nodes. All nodes are available at the
namespace `math`.


### AccessorNode

Construction:

```
new AccessorNode(object: Node, index: IndexNode)
new AccessorNode(object: Node, index: IndexNode, optionalChaining: boolean)
```

An optional property `optionalChaining` can be provided whether the accessor was 
written as optional-chaining using `a?.b`, or `a?.["b"]` with bracket notation. 
Default value is `false`. Forces evaluate to undefined if the given object is 
undefined or null.

Properties:

- `object: Node`
- `index: IndexNode`
- `name: string` (read-only) The function or method name. Returns an empty string when undefined.
- `optionalChaining: boolean`

Examples:

```js
const node1 = math.parse('a[3]')

const object = new math.SymbolNode('a')
const constant3 = new math.ConstantNode(3)
const index = new math.IndexNode([constant3])
const node2 = new math.AccessorNode(object, index)
```


### ArrayNode

Construction:

```
new ArrayNode(items: Node[])
```

Properties:

- `items: Node[]`

Examples:

```js
const node1 = math.parse('[1, 2, 3]')

const one    = new math.ConstantNode(1)
const two    = new math.ConstantNode(2)
const three  = new math.ConstantNode(3)
const node2  = new math.ArrayNode([one, two, three])
```


### AssignmentNode

Construction:

```
new AssignmentNode(object: SymbolNode, value: Node)
new AssignmentNode(object: SymbolNode | AccessorNode, index: IndexNode, value: Node)
```

Properties:

- `object: SymbolNode | AccessorNode`
- `index: IndexNode | null`
- `value: Node`
- `name: string` (read-only) The function or method name. Returns an empty string when undefined.

Examples:

```js
const node1 = math.parse('a = 3')

const object = new math.SymbolNode('a')
const value = new math.ConstantNode(3)
const node2 = new math.AssignmentNode(object, value)
```


### BlockNode

A `BlockNode` is created when parsing a multi line expression like `a=2;b=3` or
`a=2\nb=3`. Evaluating a `BlockNode` returns a `ResultSet`. The results can be
retrieved via `ResultSet.entries` or `ResultSet.valueOf()`, which contains
an `Array` with the results of the visible lines (i.e. lines not ending with
a semicolon).

Construction:

```
block = new BlockNode(Array.<{node: Node} | {node: Node, visible: boolean}>)
```

Properties:

- `blocks: Array.<{node: Node, visible: boolean}>`

Examples:

```js
const block1 = math.parse('a=1; b=2; c=3')

const a = new math.SymbolNode('a')
const one = new math.ConstantNode(1)
const ass1 = new math.AssignmentNode(a, one)

const b = new math.SymbolNode('b')
const two = new math.ConstantNode(2)
const ass2 = new math.AssignmentNode(b, two)

const c = new math.SymbolNode('c')
const three = new math.ConstantNode(3)
const ass3 = new math.AssignmentNode(c, three)

const block2 = new BlockNode([
  {node: ass1, visible: false},
  {node: ass2, visible: false},
  {node: ass3, visible: true}
])
```


### ConditionalNode

Construction:

```
new ConditionalNode(condition: Node, trueExpr: Node, falseExpr: Node)
```

Properties:

- `condition: Node`
- `trueExpr: Node`
- `falseExpr: Node`

Examples:

```js
const node1 = math.parse('a > 0 ? a : -a')

const a         = new math.SymbolNode('a')
const zero      = new math.ConstantNode(0)
const condition = new math.OperatorNode('>', 'larger', [a, zero])
const trueExpr  = a
const falseExpr = new math.OperatorNode('-', 'unaryMinus', [a])
const node2     = new math.ConditionalNode(condition, trueExpr, falseExpr)
```

### ConstantNode

Construction:

```
new ConstantNode(value: *)
```

Properties:

- `value: *`

Examples:

```js
const node1 = math.parse('2.4')

const node2 = new math.ConstantNode(2.4)
const node3 = new math.ConstantNode('foo')
```


### FunctionAssignmentNode

Construction:

```
new FunctionAssignmentNode(name: string, params: string[], expr: Node)
```

Properties:

- `name: string`
- `params: string[]`
- `expr: Node`

Examples:

```js
const node1 = math.parse('f(x) = x^2')

const x      = new math.SymbolNode('x')
const two    = new math.ConstantNode(2)
const expr   = new math.OperatorNode('^', 'pow', [x, 2])
const node2  = new math.FunctionAssignmentNode('f', ['x'], expr)
```


### FunctionNode

Construction:

```
new FunctionNode(fn: Node | string, args: Node[])
```

Properties:

- `fn: Node | string` (read-only) The object or function name which to invoke.
- `args: Node[]`

Static functions:

- `FunctionNode.onUndefinedFunction(name: string)`. This function is invoked when an undefined function is evaluated. By default, the function throws an exception "Undefined function x". The function can be overwritten to customize this behavior. See also `SymbolNode.onUndefinedSymbol`.


Examples:

```js
const node1 = math.parse('sqrt(4)')

const four  = new math.ConstantNode(4)
const node3 = new math.FunctionNode(new SymbolNode('sqrt'), [four])
```


### IndexNode

Construction:

```
new IndexNode(dimensions: Node[])
new IndexNode(dimensions: Node[], dotNotation: boolean)
```

Each dimension can be a single value, a range, or a property. The values of
indices are one-based, including range end.

An optional property `dotNotation` can be provided describing whether this index
was written using dot notation like `a.b`, or using bracket notation
like `a["b"]`. Default value is `false`. This information is used when
stringifying the IndexNode.

Properties:

- `dimensions: Node[]`
- `dotNotation: boolean`

Examples:

```js
const node1 = math.parse('A[1:3, 2]')

const A     = new math.SymbolNode('A')
const one   = new math.ConstantNode(1)
const two   = new math.ConstantNode(2)
const three = new math.ConstantNode(3)

const range = new math.RangeNode(one, three)
const index = new math.IndexNode([range, two])
const node2 = new math.AccessorNode(A, index)
```

### ObjectNode

Construction:

```
new ObjectNode(properties: Object.<string, Node>)
```

Properties:

- `properties: Object.<string, Node>`

Examples:

```js
const node1 = math.parse('{a: 1, b: 2, c: 3}')

const a = new math.ConstantNode(1)
const b = new math.ConstantNode(2)
const c = new math.ConstantNode(3)
const node2 = new math.ObjectNode({a: a, b: b, c: c})
```


### OperatorNode

Construction:

```
new OperatorNode(op: string, fn: string, args: Node[], implicit: boolean = false)
```

Additional methods:

-   `isUnary() : boolean`

    Returns true when the `OperatorNode` contains exactly one argument,
    like with a unary minus:

    ```js
    const a = new math.ConstantNode(2)
    const b = new math.OperatorNode('-', 'unaryMinus', [a])
    b.isUnary() // true
    ```

-   `isBinary() : boolean`

    Returns true when the `OperatorNode` contains exactly two arguments,
    like with most regular operators:

    ```js
    const a = new math.ConstantNode(2)
    const b = new math.ConstantNode(3)
    const c = new math.OperatorNode('+', 'add', [a, b])
    c.isBinary() // true
    ```

Properties:

- `op: string`
- `fn: string`
- `args: Node[]`
- `implicit: boolean` True in case of an implicit multiplication, false otherwise

Examples:

```js
const node1 = math.parse('2.3 + 5')

const a     = new math.ConstantNode(2.3)
const b     = new math.ConstantNode(5)
const node2 = new math.OperatorNode('+', 'add', [a, b])
```

### ParenthesisNode

Construction:

```
new ParenthesisNode(content: Node)
```

Properties:

- `content: Node`

Examples:

```js
const node1 = math.parse('(1)')

const a     = new math.ConstantNode(1)
const node2 = new math.ParenthesisNode(a)
```

### RangeNode

Construction:

```
new RangeNode(start: Node, end: Node [, step: Node])
```

Properties:

- `start: Node`
- `end: Node`
- `step: Node | null`

Examples:

```js
const node1 = math.parse('1:10')
const node2 = math.parse('0:2:10')

const zero = new math.ConstantNode(0)
const one = new math.ConstantNode(1)
const two = new math.ConstantNode(2)
const ten = new math.ConstantNode(10)

const node3 = new math.RangeNode(one, ten)
const node4 = new math.RangeNode(zero, ten, two)
```

### RelationalNode

Construction:

```
new RelationalNode(conditionals: string[], params: Node[])
```

`conditionals` is an array of strings, each of which may be 'smaller', 'larger', 'smallerEq', 'largerEq', 'equal', or 'unequal'. The `conditionals` array must contain exactly one fewer item than `params`.

Properties:

- `conditionals: string[]`
- `params: Node[]`

A `RelationalNode` efficiently represents a chained conditional expression with two or more comparison operators, such as `10 < x <= 50`. The expression is equivalent to `10 < x and x <= 50`, except that `x` is evaluated only once, and evaluation stops (is "short-circuited") once any condition tests false. Operators that are subject to chaining are `<`, `>`, `<=`, `>=`, `==`, and `!=`. For backward compatibility, `math.parse` will return an `OperatorNode` if only a single conditional is present (such as `x > 2`).

Examples:

```js

const ten = new math.ConstantNode(10)
const x = new math.SymbolNode('x')
const fifty = new math.ConstantNode(50)

const node1 = new math.RelationalNode(['smaller', 'smallerEq'], [ten, x, fifty])
const node2 = math.parse('10 < x <= 50')
```


### SymbolNode

Construction:

```
new SymbolNode(name: string)
```

Properties:

- `name: string`

Static functions:

- `SymbolNode.onUndefinedSymbol(name: string)`. This function is invoked when an undefined symbol is evaluated. By default, the function throws an exception "Undefined symbol x". The function can be overwritten to customize this behavior. See also `FunctionNode.onUndefinedFunction`.


Examples:

```js
const node = math.parse('x')

const x = new math.SymbolNode('x')
```