Compaction for two models of logarithmic-depth trees: Analysis and Experiments

In this paper we are interested in the quantitative analysis of the compaction ratio for two classical families of trees: recursive trees and plane binary increasing trees. These families are typical representatives of tree models with a small depth. More formally, asymptotically, for a random tree with $n$ nodes, its depth is of order $\ln n$. Once a tree of size $n$ is compacted by keeping only one occurrence of all fringe subtrees appearing in the tree the resulting graph contains only $O(n / \ln n)$ nodes. This result must be compared to classical results of compaction in the families of simply generated trees, where the analogous result states that the compacted structure is of size of order $n / \sqrt{\ln n}$. The result about the plane binary increasing trees has already been proved, but we propose a new and generic approach to get the result in this paper. We end the paper with an experimental quantitative study, based on a prototype implementation of compacted binary search trees, that are modeled by plane binary increasing trees.