Bounding the number of reticulation events for displaying multiple trees in a phylogenetic network

Reconstructing a parsimonious phylogenetic network that displays multiple phylogenetic trees is an important problem in phylogenetics, where the complexity of the inferred networks is measured by reticulation numbers. The reticulation number for a set of trees is defined as the minimum number of reticulations in a phylogenetic network that displays those trees. A mathematical problem is bounding the reticulation number for multiple trees over a fixed number of taxa. While this problem has been extensively studied for two trees, much less is known about the upper bounds on the reticulation numbers for three or more arbitrary trees. In this paper, we present a few non-trivial upper bounds on reticulation numbers for three or more trees.