Directed and undirected trees

In both graph theory and order theory there is a concept of trees. In graph theory there an undirected trees and in order theory there are directed trees, which are given some root. A tree graph is shown below, its lack of direction is because it is a graph.



One interesting property of graphs is that they have a set of metric intervals associated with them, these metric intervals contain all points between any two points. When the graph is not traceable, then the set of metric intervals will not be upper bounded like it was with the path graph considered earlier. Well the metric intervals of the path graph form a triangular structure, only the maximal sets of the metric intervals of the tree form a triangular structure among there parts in the more general case as shown below.



Well that set of intervals of the tree graph can be considerably larger then the original graph (to a roughly triangular quadratic extent) it is interesting to consider how we can go from the original tree graph to a directed tree graph on the same number of vertices. In order to convert an undirected tree to a directed tree we need some special node to be the root. I realized that the directed version of a tree at some root is simply the set of intervals of the tree with the root as one of its endpoints. This turns into a lower tree with the root as the minimal element, as shown below for the case of the tree graph with the root chosen at zero.



Each set of intervals produced from some root produces a different directed version of the tree. The total set of intervals of the tree is therefore equal to the union of the different rooted orientations of the tree produced in this manner and the empty interval, with non trivial intervals repeated. This rooted set of intervals is also an order containment family. This means that the set of metric intervals centered around some point is also equal to the set of the principal ideals of the corresponding rooted tree.