Archive for June 2011

## A third view on trees

A few recent posts have played with trees from two perspectives. The more commonly used I call "top-down", because the top-level structure is most immediately apparent. A top-down binary tree is either a leaf or a pair of such trees, and that pair can be accessed without wading through intervening structure. Much less commonly used are "bottom-up" trees. A bottom-up binary tree is either a leaf or a single such tree of pairs. In the non-leaf case, the pair structure of the tree elements is accessible by operations like mapping, folding, or scanning. The difference is between a pair of trees and a tree of pairs.

As an alternative to the top-down and bottom-up views on trees, I now want to examine a third view, which is a hybrid of the two. Instead of pairs of trees or trees of pairs, this hybrid view is of trees of trees, and more specifically of bottom-up trees of top-down trees. As we’ll see, these hybrid trees emerge naturally from the top-down and bottom-up views. A later post will show how this third view lends itself to an in-place (destructive) scan algorithm, suitable for execution on modern GPUs.

Edits:

• 2011-06-04: "Suppose we have a bottom-up tree of top-down trees, i.e., `t ∷ TB (TT a)`. Was backwards. (Thanks to Noah Easterly.)
• 2011-06-04: Notation: "`f ➶ n`" and "`f ➴ n`".

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