How to Grow an Optimal Tree

As an effort to show something that may be of broad interest, yet not to repeat what I have done in the interview, we choose this topic on trees, one of the fundamental and elementary subjects in Graph Theory. Various concepts of trees related to distances, subtrees, etc. will be studied, originated from different fields in pure Graph Theory, Bio Chemistry and Phylogeny. Following the introduction of motivations and definitions, the extremal (optimal) tree in each category will be discussed, we present the ideas of different ways to achieve these extremal trees and the somewhat mysterious bound between these techniques/concepts.