# 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:

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 build by manually creating nodes:

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:

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:

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:

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:

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:

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. Information about the options in Customization.

• toString(options: object) : string

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

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


Information about the options in Customization.

• toTex(options: object): string

Get a LaTeX representation of the expression. Example:

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


Information about the options in Customization.

• 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:

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:

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)


Properties:

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

Examples:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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)


• isUnary() : boolean

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

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:

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:

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:

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:

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:


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:

const node = math.parse('x')

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