Yesterday, I asked about the basic AVL cost and explained the tower-of-Hanoi cost structure of a very simple pre-re-balancing AVL algorithm in Stackexchange. Today, I found my explanation deleted. (Almost) everybody knows about the very simple exponential tower-of-Hanoi cost structure, and you can naturally guess that similar structures are found in many other algorithms. The symmetric groups are non-comutative; there is no simple isomorphism between the original sorting and the logarithmic tree swapping (assuming the mapping volume is logarithmic); a simple substitution generates a massive tree perturbation. Because I knew less about AVL tree than red-black tree, my first (simple) normalization of the AVL algorithm was {left,even,right} ~ {down,0,up} ~ {green,green,red} , which is a coarse-grained 2-adic rounding. In order to deterministically clear one red-light (excluding the simple subtree costs), you must first move down one new green-light from the tree-top, which is generated by a simple rotation. In order to move down one green-light, you must clear every red-light one by one. The node doesn't move; only the green bits move (just as the pixels of the PC screen flow.) The (partial) cost of the AVL algorithm (excluding the subtree cost) is (roughly) summerized as dF(y)/dy> ∫F(y)dy; in other words, partial cost=exp(height); this is a very simple exponential cost structure with the randomized bad-balance height. Because the green pixels move down, the red pixels move up; it resembles the global red-waterfall of the red-black tree; the red pixel flow direction is the same. The tree-top of a tree algorithm always works as a very simple heatbath. Because the red-green distribution is randomly flipped after every node insertion, the tree is in a hot equilibrium state. The red-entropy swap-out is (super-) linear to time.