--- title: "TreeForm" language: "en" type: "Symbol" summary: "TreeForm[expr] displays expr as a tree with different levels at different depths. TreeForm[expr, n] displays expr as a tree only down to level n." keywords: - pointer structure - tree formatted expression - tree formatting - tree structure - binary tree - kary tree - expression tree - formula tree canonical_url: "https://reference.wolfram.com/language/ref/TreeForm.html" source: "Wolfram Language Documentation" related_guides: - title: "Expression Structure" link: "https://reference.wolfram.com/language/guide/ExpressionStructure.en.md" - title: "Expressions" link: "https://reference.wolfram.com/language/guide/Expressions.en.md" - title: "Tree Visualization" link: "https://reference.wolfram.com/language/guide/TreeVisualization.en.md" - title: "Graph Visualization" link: "https://reference.wolfram.com/language/guide/GraphVisualization.en.md" - title: "Trees" link: "https://reference.wolfram.com/language/guide/Trees.en.md" related_workflows: - title: "Find the Underlying Tree Structure of an Expression" link: "https://reference.wolfram.com/language/workflow/FindTheUnderlyingTreeStructureOfAnExpression.en.md" related_functions: - title: "FullForm" link: "https://reference.wolfram.com/language/ref/FullForm.en.md" - title: "MatrixForm" link: "https://reference.wolfram.com/language/ref/MatrixForm.en.md" - title: "LeafCount" link: "https://reference.wolfram.com/language/ref/LeafCount.en.md" - title: "ExpressionTree" link: "https://reference.wolfram.com/language/ref/ExpressionTree.en.md" - title: "TreePlot" link: "https://reference.wolfram.com/language/ref/TreePlot.en.md" - title: "TreeGraph" link: "https://reference.wolfram.com/language/ref/TreeGraph.en.md" - title: "ExpressionGraph" link: "https://reference.wolfram.com/language/ref/ExpressionGraph.en.md" - title: "Graph" link: "https://reference.wolfram.com/language/ref/Graph.en.md" related_tutorials: - title: "Expressions as Trees" link: "https://reference.wolfram.com/language/tutorial/Expressions.en.md#14609" --- # TreeForm TreeForm[expr] displays expr as a tree with different levels at different depths. TreeForm[expr, n] displays expr as a tree only down to level n. ## Details and Options * With a notebook front end, ``TreeForm`` displays as a graphical tree in ``StandardForm`` and ``TraditionalForm``. * ``TreeForm`` takes the same options as ``TreePlot``. * ``VertexLabeling -> Automatic`` does not display expression fragments at each node, but still gives subtree expressions as tooltips. * ``VertexLabeling -> False`` does not give expression fragments or tooltips. * The typeset form of ``TreeForm[expr]`` is interpreted the same as ``expr`` when used in input. » * When an input evaluates to ``TreeForm[expr]``, ``TreeForm`` does not appear in the output. » ### List of all options | | | | | ---------------------- | --------------- | ---------------------------------------------------------------------------------- | | AlignmentPoint | Center | the default point in the graphic to align with | | AspectRatio | Automatic | ratio of height to width | | Axes | False | whether to draw axes | | AxesLabel | None | axes labels | | AxesOrigin | Automatic | where axes should cross | | AxesStyle | {} | style specifications for the axes | | Background | None | background color for the plot | | BaselinePosition | Automatic | how to align with a surrounding text baseline | | BaseStyle | {} | base style specifications for the graphic | | ContentSelectable | Automatic | whether to allow contents to be selected | | CoordinatesToolOptions | Automatic | detailed behavior of the coordinates tool | | DataRange | Automatic | the range of vertex coordinates to generate | | DirectedEdges | False | whether to interpret Rule as DirectedEdge | | EdgeLabels | None | labels and placements for edges | | EdgeLabelStyle | Automatic | style to use for edge labels | | EdgeShapeFunction | Automatic | generate graphic shapes for edges | | EdgeStyle | Automatic | styles for edges | | Epilog | {} | primitives rendered after the main plot | | FormatType | TraditionalForm | the default format type for text | | Frame | False | whether to put a frame around the plot | | FrameLabel | None | frame labels | | FrameStyle | {} | style specifications for the frame | | FrameTicks | Automatic | frame ticks | | FrameTicksStyle | {} | style specifications for frame ticks | | GraphHighlight | {} | vertices and edges to highlight | | GraphHighlightStyle | Automatic | style for highlight | | GridLines | None | grid lines to draw | | GridLinesStyle | {} | style specifications for grid lines | | ImageMargins | 0. | the margins to leave around the graphic | | ImagePadding | All | what extra padding to allow for labels etc. | | ImageSize | Automatic | the absolute size at which to render the graphic | | LabelStyle | {} | style specifications for labels | | LayerSizeFunction | (1) | the height to allow for each layer | | Method | Automatic | details of graphics methods to use | | PerformanceGoal | Automatic | aspects of performance to try to optimize | | PlotLabel | None | an overall label for the plot | | PlotRange | All | range of values to include | | PlotRangeClipping | False | whether to clip at the plot range | | PlotRangePadding | Automatic | how much to pad the range of values | | PlotRegion | Automatic | the final display region to be filled | | PlotStyle | Automatic | graphics directives to determine styles | | PlotTheme | Automatic | overall theme for the graph | | PreserveImageOptions | Automatic | whether to preserve image options when displaying new versions of the same graphic | | Prolog | {} | primitives rendered before the main plot | | RotateLabel | True | whether to rotate y labels on the frame | | Ticks | Automatic | axes ticks | | TicksStyle | {} | style specifications for axes ticks | | VertexCoordinates | Automatic | coordinates for vertices | | VertexLabels | None | labels and placements for vertices | | VertexLabelStyle | Automatic | style to use for vertex labels | | VertexShape | Automatic | graphic shape for vertices | | VertexShapeFunction | Automatic | generate graphic shapes for vertices | | VertexSize | Automatic | size of vertices | | VertexStyle | Automatic | styles for vertices | --- ## Examples (37) ### Basic Examples (3) A symbolic expression formatted as a tree: ```wl In[1]:= TreeForm[a + b ^ 2 + c ^ 3 + d] Out[1]//TreeForm= TreeForm[a + b^2 + c^3 + d] ``` --- Show the tree form for the first two levels in the expression: ```wl In[1]:= p = HornerForm[1 + x + x ^ 2 + x ^ 3, x] Out[1]= 1 + x (1 + x (1 + x)) In[2]:= TreeForm[p, 2] Out[2]//TreeForm= TreeForm[1 + x*(1 + x*(1 + x)), 2] ``` --- A graphics expression formatted as a tree: ```wl In[1]:= [image]//TreeForm Out[1]//TreeForm= TreeForm[Graphics[{RGBColor[1, 0, 0], Disk[{0, 0}]}, ImageSize -> 80]] ``` ### Scope (9) A formatted symbolic expression with mathematical constants: ```wl In[1]:= TreeForm[E^n I π] Out[1]//TreeForm= TreeForm[E^((Complex[0, 1]*n)*Pi)] ``` --- A formatted expression with symbolic, exact and inexact values: ```wl In[1]:= TreeForm[(1/3) Sqrt[10]E^.25^x] Out[1]//TreeForm= TreeForm[(Rational[1, 3]*Sqrt[10])*E^0.25^x] ``` --- A nested list: ```wl In[1]:= TreeForm[{1, {2, {3}}}] Out[1]//TreeForm= TreeForm[{1, {2, {3}}}] ``` --- An expression containing subscripted variables: ```wl In[1]:= TreeForm[Subscript[x, 1] + Subscript[x, 2]] Out[1]//TreeForm= TreeForm[Subscript[x, 1] + Subscript[x, 2]] ``` --- A formatted expression with a special superscript and ``OverBar`` : ```wl In[1]:= TreeForm[x^† + Overscript[y, _]] Out[1]//TreeForm= TreeForm[OverBar[y] + SuperDagger[x]] ``` --- A graphic object: ```wl In[1]:= [image]//TreeForm Out[1]//TreeForm= TreeForm[Graphics[{RGBColor[1, 0, 0], Disk[{0, 0}]}, ImageSize -> 80]] ``` --- A ``Series`` expression: ```wl In[1]:= ser = Series[Sin[x], {x, 0, 5}] Out[1]= SeriesData[x, 0, {1, 0, Rational[-1, 6], 0, Rational[1, 120]}, 1, 6, 1] In[2]:= TreeForm[ser] Out[2]//TreeForm= TreeForm[SeriesData[x, 0, {1, 0, Rational[-1, 6], 0, Rational[1, 120]}, 1, 6, 1]] ``` --- An expression containing ``Hold`` : ```wl In[1]:= TreeForm[Hold[1 + 2 + 3]] Out[1]//TreeForm= TreeForm[Hold[1 + 2 + 3]] ``` --- Limit the levels shown: ```wl In[1]:= TreeForm[Nest[{#, {#, {#}}}&, x, 2], 4] Out[1]//TreeForm= TreeForm[{{x, {x, {x}}}, {{x, {x, {x}}}, {{x, {x, {x}}}}}}, 4] ``` ### Options (14) #### AspectRatio (2) By default a suitable aspect ratio is calculated for good visual appearance: ```wl In[1]:= TreeForm[Nest[{#, #, #}&, x, 2]] Out[1]//TreeForm= TreeForm[{{x, x, x}, {x, x, x}, {x, x, x}}] ``` --- Change the aspect ratio: ```wl In[1]:= TreeForm[Nest[{#, #, #}&, x, 2], AspectRatio -> .3] Out[1]//TreeForm= TreeForm[{{x, x, x}, {x, x, x}, {x, x, x}}, AspectRatio -> 0.3] ``` #### DirectedEdges (1) Show directed edges: ```wl In[1]:= TreeForm[Integrate[1 / (1 + 1 / (1 + x)), x], DirectedEdges -> True] Out[1]//TreeForm= TreeForm[x - Log[2 + x], DirectedEdges -> True] ``` #### EdgeRenderingFunction (2) Draw edges using blue arrows set back by 30%: ```wl In[1]:= TreeForm[D[x Exp[x], x], EdgeRenderingFunction -> ({Blue, Arrow[#, .3]}&)] Out[1]//TreeForm= TreeForm[E^x + E^x*x, EdgeRenderingFunction -> ({Blue, Arrow[#1, 0.3]} & )] ``` --- Draw vertices only: ```wl In[1]:= TreeForm[D[x Exp[x], x], EdgeRenderingFunction -> None] Out[1]//TreeForm= TreeForm[E^x + E^x*x, EdgeRenderingFunction -> None] ``` #### LayerSizeFunction (1) Draw a tree with the first level of height 1, the second level 2, etc.: ```wl In[1]:= TreeForm[Nest[{#, #}&, x, 4], LayerSizeFunction -> (#&)] Out[1]//TreeForm= TreeForm[{{{{x, x}, {x, x}}, {{x, x}, {x, x}}}, {{{x, x}, {x, x}}, {{x, x}, {x, x}}}}, LayerSizeFunction -> (#1 & )] ``` #### PlotRangePadding (1) Use different ``PlotRangePadding`` around the drawing: ```wl In[1]:= Table[Framed[TreeForm[D[x Cos[x] Sin[x], x], PlotRangePadding -> p], FrameMargins -> 0], {p, {0, Scaled[0.2]}}] Out[1]= [image] ``` #### PlotStyle (3) Specify an overall style for the drawing: ```wl In[1]:= Table[TreeForm[Cos[Exp[x]], PlotStyle -> ps], {ps, {Red, PointSize[Medium], Dashed, Directive[Red, PointSize[Medium], Dashed]}}] Out[1]= {TreeForm[Cos[E^x], PlotStyle -> RGBColor[1, 0, 0]], TreeForm[Cos[E^x], PlotStyle -> PointSize[Medium]], TreeForm[Cos[E^x], PlotStyle -> Dashing[{Small, Small}]], TreeForm[Cos[E^x], PlotStyle -> Directive[RGBColor[1, 0, 0], PointSize[Medium], Dashing[{Small, Small}]]]} ``` --- ``PlotStyle`` can be combined with ``VertexRenderingFunction``, which has higher priority: ```wl In[1]:= TreeForm[Cos[Exp[x]], PlotStyle -> Directive[PointSize[Large], Red], VertexRenderingFunction -> Function[{p, l}, {Green, Point[p]}]] Out[1]//TreeForm= TreeForm[Cos[E^x], PlotStyle -> Directive[PointSize[Large], RGBColor[1, 0, 0]], VertexRenderingFunction -> Function[{p, l}, {Green, Point[p]}]] ``` --- ``PlotStyle`` can be combined with ``EdgeRenderingFunction``, which has higher priority: ```wl In[1]:= TreeForm[Cos[Exp[x]], PlotStyle -> Directive[Dashed, Red], EdgeRenderingFunction -> Function[{p, vl, el}, {Green, Line[p]}]] Out[1]//TreeForm= TreeForm[Cos[E^x], PlotStyle -> Directive[Dashing[{Small, Small}], RGBColor[1, 0, 0]], EdgeRenderingFunction -> Function[{p, vl, el}, {Green, Line[p]}]] ``` #### VertexLabeling (1) Display subtree expressions as tooltips: ```wl In[1]:= TreeForm[Nest[{#, {#}, #}&, x, 3], VertexLabeling -> Automatic] Out[1]//TreeForm= TreeForm[{{{x, {x}, x}, {{x, {x}, x}}, {x, {x}, x}}, {{{x, {x}, x}, {{x, {x}, x}}, {x, {x}, x}}}, {{x, {x}, x}, {{x, {x}, x}}, {x, {x}, x}}}, VertexLabeling -> Automatic] ``` #### VertexRenderingFunction (2) Show no vertices: ```wl In[1]:= TreeForm[D[Sin[Cos[x]], x], VertexRenderingFunction -> None] Out[1]//TreeForm= TreeForm[(-Cos[Cos[x]])*Sin[x], VertexRenderingFunction -> None] ``` --- Render vertices using a predefined graphic: ```wl In[1]:= gr = [image]; In[2]:= TreeForm[Nest[{#, {#}}&, x, 3], VertexRenderingFunction -> (Inset[gr, #1]&)] Out[2]//TreeForm= TreeForm[{{{x, {x}}, {{x, {x}}}}, {{{x, {x}}, {{x, {x}}}}}}, VertexRenderingFunction -> (Inset[gr, #1] & )] ``` #### VertexCoordinateRules (1) Explicitly specify all vertex coordinates: ```wl In[1]:= TreeForm[x ^ 2 + x ^ 4, VertexCoordinateRules -> {{0, 2}, {0, 1}, {0, 0}, {2, -1}, {2, 1}, {1, 3}, {3, 1}}] Out[1]//TreeForm= TreeForm[x^2 + x^4, VertexCoordinateRules -> {{0, 2}, {0, 1}, {0, 0}, {2, -1}, {2, 1}, {1, 3}, {3, 1}}] ``` ### Applications (1) Visualize the Wolfram Axiom for Boolean algebra as a tree: ```wl In[1]:= TreeForm[f[f[f[p, q], r], f[p, f[f[p, r], p]]] == r] Out[1]//TreeForm= TreeForm[f[f[f[p, q], r], f[p, f[f[p, r], p]]] == r] ``` ### Properties & Relations (6) ``FullForm`` gives a linear expression similar to ``TreeForm`` : ```wl In[1]:= FullForm[1 / (a + b) * x ^ 2] Out[1]//FullForm= Times[Power[Plus[a, b], -1], Power[x, 2]] In[2]:= TreeForm[1 / (a + b) * x ^ 2] Out[2]//TreeForm= TreeForm[x^2/(a + b)] ``` ``OutputForm`` of ``TreeForm`` gives a textual display of the expression: ```wl In[3]:= TreeForm[1 / (a + b) * x ^ 2]//OutputForm ``` Out[3]//OutputForm= x^2/(a + b) --- Use ``TreePlot`` to plot a tree graph: ```wl In[1]:= TreePlot[Table[i -> RandomInteger[{0, i - 1}], {i, 50}]] Out[1]= [image] ``` --- Use ``GraphPlot`` or ``GraphPlot3D`` for general undirected graphs: ```wl In[1]:= GraphPlot[Table[i -> Mod[i ^ 2, 150], {i, 0, 150}]] Out[1]= [image] In[2]:= GraphPlot3D[Table[i -> Mod[i ^ 2, 150], {i, 0, 150}]] Out[2]= [image] ``` --- Use ``LayeredGraphPlot`` for hierarchical-style drawing of directed graphs: ```wl In[1]:= LayeredGraphPlot[Table[RandomInteger[{0, 50}] -> i, {i, 50}]] Out[1]= [image] ``` --- The typeset form of ``TreeForm[expr]`` is interpreted the same as ``expr`` when used in input: ```wl In[1]:= {TreeForm[x ^ 2]} Out[1]= {TreeForm[x^2]} ``` Copy the output and paste it into an input cell. The ``TreeForm[x^2]`` is interpreted as ``x^2`` : ```wl In[2]:= {TreeForm[x^2]} Out[2]= {x^2} ``` --- When an input evaluates to ``TreeForm[expr]``, ``TreeForm`` does not appear in the output: ```wl In[1]:= TreeForm[x ^ 2] Out[1]//TreeForm= TreeForm[x^2] ``` ``Out`` is assigned the value ``x^2``, not ``TreeForm[x ^ 2]`` : ```wl In[2]:= % Out[2]= x^2 ``` ### Possible Issues (2) Compound heads are not laid out as trees: ```wl In[1]:= TreeForm[g[a][b]] Out[1]//TreeForm= TreeForm[g[a][b]] ``` --- Even when an output omits ``TreeForm`` from the top level, it is not stripped from subexpressions: ```wl In[1]:= e = TreeForm[x ^ 2] Out[1]//TreeForm= TreeForm[x^2] ``` The output does not have ``TreeForm`` in it: ```wl In[2]:= % Out[2]= x^2 ``` However, the variable ``e`` does have ``TreeForm`` in it, which may affect subsequent evaluations: ```wl In[3]:= FullForm[e] Out[3]//FullForm= TreeForm[Power[x, 2]] ``` The product is not evaluated due to the intervening ``TreeForm`` : ```wl In[4]:= x * e Out[4]= x TreeForm[x^2] ``` Assign variables first and then apply ``TreeForm`` to the result to maintain computability: ```wl In[5]:= (f = x ^ 2)//TreeForm Out[5]//TreeForm= TreeForm[x^2] In[6]:= x * f Out[6]= x^3 ``` ### Neat Examples (2) A complete binary tree: ```wl In[1]:= Nest[{#, #}&, x, 4] Out[1]= {{{{x, x}, {x, x}}, {{x, x}, {x, x}}}, {{{x, x}, {x, x}}, {{x, x}, {x, x}}}} In[2]:= TreeForm[%] Out[2]//TreeForm= TreeForm[{{{{x, x}, {x, x}}, {{x, x}, {x, x}}}, {{{x, x}, {x, x}}, {{x, x}, {x, x}}}}] ``` A complete ternary tree: ```wl In[3]:= Nest[{#, #, #}&, x, 3] Out[3]= {{{x, x, x}, {x, x, x}, {x, x, x}}, {{x, x, x}, {x, x, x}, {x, x, x}}, {{x, x, x}, {x, x, x}, {x, x, x}}} In[4]:= TreeForm[%] Out[4]//TreeForm= TreeForm[{{{x, x, x}, {x, x, x}, {x, x, x}}, {{x, x, x}, {x, x, x}, {x, x, x}}, {{x, x, x}, {x, x, x}, {x, x, x}}}] ``` --- A symmetric tree: ```wl In[1]:= TreeForm[Nest[{#, {#}, #}&, x, 5], VertexLabeling -> Automatic] Out[1]//TreeForm= TreeForm[{{{{{x, {x}, x}, {{x, {x}, x}}, {x, {x}, x}}, {{{x, {x}, x}, {{x, {x}, x}}, {x, {x}, x}}}, {{x, {x}, x}, {{x, {x}, x}}, {x, {x}, x}}}, {{{{x, {x}, x}, {{x, {x}, x}}, {x, {x}, x}}, {{{x, {x}, x}, {{x, {x}, x}}, {x, {x}, x}}}, {{x ... {{x, {x}, x}}, {x, {x}, x}}}, {{x, {x}, x}, {{x, {x}, x}}, {x, {x}, x}}}}, {{{x, {x}, x}, {{x, {x}, x}}, {x, {x}, x}}, {{{x, {x}, x}, {{x, {x}, x}}, {x, {x}, x}}}, {{x, {x}, x}, {{x, {x}, x}}, {x, {x}, x}}}}}, VertexLabeling -> Automatic] ``` An asymmetric tree: ```wl In[2]:= TreeForm[Nest[{{#}, #, #}&, x, 5], VertexLabeling -> Automatic] Out[2]//TreeForm= TreeForm[{{{{{{{{{{x}, x, x}}, {{x}, x, x}, {{x}, x, x}}}, {{{{x}, x, x}}, {{x}, x, x}, {{x}, x, x}}, {{{{x}, x, x}}, {{x}, x, x}, {{x}, x, x}}}}, {{{{{{x}, x, x}}, {{x}, x, x}, {{x}, x, x}}}, {{{{x}, x, x}}, {{x}, x, x}, {{x}, x, x}}, ... }}, {{x}, x, x}, {{x}, x, x}}, {{{{x}, x, x}}, {{x}, x, x}, {{x}, x, x}}}, {{{{{{x}, x, x}}, {{x}, x, x}, {{x}, x, x}}}, {{{{x}, x, x}}, {{x}, x, x}, {{x}, x, x}}, {{{{x}, x, x}}, {{x}, x, x}, {{x}, x, x}}}}}, VertexLabeling -> Automatic] ``` ## See Also * [`FullForm`](https://reference.wolfram.com/language/ref/FullForm.en.md) * [`MatrixForm`](https://reference.wolfram.com/language/ref/MatrixForm.en.md) * [`LeafCount`](https://reference.wolfram.com/language/ref/LeafCount.en.md) * [`ExpressionTree`](https://reference.wolfram.com/language/ref/ExpressionTree.en.md) * [`TreePlot`](https://reference.wolfram.com/language/ref/TreePlot.en.md) * [`TreeGraph`](https://reference.wolfram.com/language/ref/TreeGraph.en.md) * [`ExpressionGraph`](https://reference.wolfram.com/language/ref/ExpressionGraph.en.md) * [`Graph`](https://reference.wolfram.com/language/ref/Graph.en.md) ## Tech Notes * [Expressions as Trees](https://reference.wolfram.com/language/tutorial/Expressions.en.md#14609) ## Related Guides * [Expression Structure](https://reference.wolfram.com/language/guide/ExpressionStructure.en.md) * [`Expressions`](https://reference.wolfram.com/language/guide/Expressions.en.md) * [Tree Visualization](https://reference.wolfram.com/language/guide/TreeVisualization.en.md) * [Graph Visualization](https://reference.wolfram.com/language/guide/GraphVisualization.en.md) * [`Trees`](https://reference.wolfram.com/language/guide/Trees.en.md) ## Related Workflows * [Find the Underlying Tree Structure of an Expression](https://reference.wolfram.com/language/workflow/FindTheUnderlyingTreeStructureOfAnExpression.en.md) ## Related Links * [An Elementary Introduction to the Wolfram Language: Expressions and Their Structure](https://www.wolfram.com/language/elementary-introduction/33-expressions-and-their-structure.html) ## History * Introduced in 1988 (1.0) \| [Updated in 2007 (6.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn60.en.md)