Structural theory of trees. II. Completeness and completions of trees

Trees are partial orderings where every element has a linearly ordered set of smaller elements. We define and study several natural notions of completeness of trees, extending Dedekind completeness of linear orders and Dedekind-MacNeille completions of partial orders. We then define constructions of \emph{tree completions} that extend any tree to a minimal one satisfying the respective completeness property.