Decision Trees of Algorithms and a Semivaluation to Measure Their Distance

We use the set T_n of binary trees with n leaves to study decision trees of algorithms. The set T_n of binary trees with n leaves can be ordered by the so called "imbalance" order, where two trees are related in the order iff the second is less "balanced" than the first. This order forms a lattice. We show that this lattice is nonmodular and extend the imbalance lattice with an algebraic operation. The operation corresponds to the extension of a binary tree with new binary trees at the leafs, which reflects the effect of recursive calls in an algorithm on the decision tree and we will characterize as an illustration the decision tree of the insertion sort algorithm. We investigate the semivaluations on the binary trees which is related to the running time of the algorithm.