A Class of Trees Having Near-Best Balance

TL;DR

This paper introduces a new class of binary trees that achieve near-optimal balance for computational efficiency, offering flexibility over traditional divide-and-conquer trees in various calculation contexts.

Contribution

The paper presents a novel class of trees based on the Colless index that are easy to construct and provide more flexible grouping and ordering options for calculations.

Findings

Trees are constructed from binary decomposition of term counts.

They achieve near-best balance in terms of the Colless index.

Flexible grouping allows for efficiency trade-offs.

Abstract

Full binary trees naturally represent commutative non-associative products. There are many important examples of these products: finite-precision floating-point addition and NAND gates, among others. Balance in such a tree is highly desirable for efficiency in calculation. The best balance is attained with a divide-and-conquer approach. However, this may not be the optimal solution, since the success of many calculations is dependent on the grouping and ordering of the calculation, for reasons ranging from the avoidance of rounding error, to calculating with varying precision, to the placement of calculation within a heterogeneous system. We introduce a new class of computational trees having near-best balance in terms of the Colless index from mathematical phylogenetics. These trees are easily constructed from the binary decomposition of the number of terms in the problem. They also permit much more flexibility than the optimally balanced divide-and-conquer trees. This gives needed freedom in the grouping and ordering of calculation, and allows intelligent efficiency trade-offs.