Additional Catalan Transformations

The following depicts many Catalan problem transformations. Personally, I find it very hard to follow written descriptions of these transformations, so instead hope to describe these conversions with an image.

First we see a binary tree, then we see how a binary tree can be converted to a set of parentheses. That parenthesis string can then be converted to a ballot sequence, or a mountain range with equal up and down strokes, or a lattice path, or a stair step tiling. The long story short, all of these structures are equivalent, and they use Catalan numbers as an almost "implicit" data structure.