Decision Mesh

Freeform triangles. Every edge carries a candidate vertex at its midpoint; activating it splits the one or two triangles on that edge through their opposite corners. Any edge can be cut, so the mesh can turn to follow a feature in any direction, at the price of thin triangles, which the aspect limit keeps in check.

Right triangles. Every triangle is right isosceles and is only ever cut through the midpoint of its hypotenuse, which gives two smaller right isosceles triangles. If the triangle across that hypotenuse would be left with a vertex hanging in its side, it is cut first (the completion step), so the mesh never has hanging vertices and its triangles never get thin, however long it grows. It cannot turn to follow a feature; it can only get finer where the feature is.

Rectangles. Every cell can be cut across either axis, so the refinement can follow anisotropy that lines up with the axes; inside a cell the fit is bilinear, which carries the interaction of the two coordinates that a flat triangle cannot. A vertex left hanging in the middle of a coarser cell's edge takes that edge's endpoints as its parents, which keeps the surface continuous with no constraints to store.

In the right triangles and the rectangles, a vertex's height is its parents' average plus a surplus of its own, and a new vertex costs a parameter only once the rule frees its surplus. In all three, each candidate knows the least-squares change it would make with everything else held fixed and how much squared error that would save; each step takes the largest saving. The tree is ordinary best-first CART. All four are grown to the same number of free parameters: one height or surplus per free vertex, one mean per tree leaf.