Last year we released Version 13.0 of the Wolfram Language. Here are the updates in trees since then, including the latest features in 13.1.

 

Trees Continue to Grow 🌱🌳

In Version 12.3 we introduced Tree as a new fundamental construct in the Wolfram Language. In Version 13.0 we added a variety of styling options for trees, and in Version 13.1 we’re adding more styling as well as a variety of new fundamental features.

An important update to the fundamental Tree construct in Version 13.1 is the ability to name branches at each node, by giving them in an association:

&#10005

All tree functions now include support for associations:

&#10005

In many uses of trees the labels of nodes are crucial. But particularly in more abstract applications one often wants to deal with unlabeled trees. In Version 13.1 the function UnlabeledTree (roughly analogously to UndirectedGraph) takes a labeled tree, and basically removes all visible labels. Here is a standard labeled tree

&#10005

and here’s the unlabeled analog:

&#10005

In Version 12.3 we introduced ExpressionTree for deriving trees from general symbolic expressions. Our plan is to have a wide range of “special trees” appropriate for representing different specific kinds of symbolic expressions. We’re beginning this process in Version 13.1 by, for example, having the concept of “Dataset trees”. Here’s ExpressionTree converting a dataset to a tree:

&#10005

And now here’s TreeExpression “inverting” that, and producing a dataset:

&#10005

(Remember the convention that *Tree functions return a tree; while Tree* functions take a tree and return something else.)

Here’s a “graph rendering” of a more complicated dataset tree:

&#10005

The new function TreeLeafCount lets you count the total number of leaf nodes on a tree (basically the analog of LeafCount for a general symbolic expression):

&#10005

Another new function in Version 13.1 that’s often useful in getting a sense of the structure of a tree without inspecting every node is RootTree. Here’s a random tree:

&#10005

RootTree can get a subtree that’s “close to the root”:

&#10005

It can also get a subtree that’s “far from the leaves”, in this case going down to elements that are at level –2 in the tree:

&#10005

In some ways the styling of trees is like the styling of graphs—though there are some significant differences as a result of the hierarchical nature of trees. By default, options inserted into a particular tree element affect only that tree element:

&#10005

But you can give rules that specify how elements in the subtree below that element are affected:

&#10005

In Version 13.1 there is now detailed control available for styling both nodes and edges in the tree. Here’s an example that gives styling for parent edges of nodes:

&#10005

Options like TreeElementStyle determine styling from the positions of elements. TreeElementStyleFunction, on the other hand, determines styling by applying a function to the data at each node:

&#10005

This uses both data and position information for each node:

&#10005

In analogy with VertexShapeFunction for graphs, TreeElementShapeFunction provides a general mechanism to specify how nodes of a tree should be rendered. This named setting for TreeElementShapeFunction makes every node be displayed as a circle:

&#10005

Last year we released Version 13.0 of the Wolfram Language. Here are the updates in visual effects and beautification since then, including the latest features in 13.1.

 

Visual Effects & Beautification

At first it seemed like a minor feature. But once we’d implemented it, we realized it was much more useful than we’d expected. Just as you can style a graphics object with its color (and, as of Version 13.0, its filling pattern), now in Version 13.1 you can style it with its drop shadowing:

&#10005

Drop shadowing turns out to be a nice way to “bring graphics to life”

&#10005

or to emphasize one element over others:

&#10005

It works well in geo graphics as well:

&#10005

DropShadowing allows detailed control over the shadows: what direction they’re in, how blurred they are and what color they are:

&#10005

Drop shadowing is more complicated “under the hood” than one might imagine. And when possible it actually works using hardware GPU pixel shaders—the same technology that we’ve used since Version 12.3 to implement material-based surface textures for 3D graphics. In Version 13.1 we’ve explicitly exposed some well-known underlying types of 3D shading. Here’s a geodesic polyhedron (yes, that’s another new function in Version 13.1), with its surface normals added (using the again new function EstimatedPointNormals):

&#10005

Here’s the most basic form of shading: flat shading of each facet (and the specularity in this case doesn’t “catch” any facets):

&#10005

Here now is Gouraud shading, with a somewhat-faceted glint:

&#10005

And then there’s Phong shading, looking somewhat more natural for a sphere:

&#10005

Ever since Version 1.0, we’ve had an interactive way to rotate—and zoom into—3D graphics. (Yes, the mechanism was a bit primitive 34 years ago, but it rapidly got to more or less its modern form.) But in Version 13.1 we’re adding something new: the ability to “dolly” into a 3D graphic, imitating what would happen if you actually walked into a physical version of the graphic, as opposed to just zooming your camera:

&#10005

And, yes, things can get a bit surreal (or “treky”)—here dollying in and then zooming out: