Revealing the Building Blocks of Tree Balance: Fundamental Units of the Sackin and Colless Indices

Abstract (Im)balance indices can be used to quantify the (im)balance of trees by assigning numerical scores to them. An easy way to generate a new index is to construct a compound index, e.g., a linear combination of established indices. Two of the most prominent and widely used imbalance indices are the Sackin index and the Colless index. In this study, we show that these classic indices are themselves compound in nature: they can be decomposed into more elementary components that independently satisfy the defining properties of a tree (im)balance index. We further show that the difference Colless minus Sackin results in another imbalance index that is minimized (among others) by all Colless minimal trees. Conversely, the difference Sackin minus Colless forms a balance index. Finally, we compare the building blocks of which the Sackin and the Colless indices consist to these indices as well as to the stairs2 index, which is another index from the literature. Our results suggest that the elementary building blocks we identify are not only foundational to established indices but also valuable tools for analyzing disagreement among indices when comparing the balance of different trees. Along the way, we investigate the so-called echelon tree, which plays an important role for several (im)balance indices, and present the first non-recursive algorithm to construct it.