thumb|A polytree

In mathematics, and more specifically in graph theory, a polytree (also called directed tree, oriented tree or singly connected network) is a directed acyclic graph whose underlying undirected graph is a tree. In other words, a polytree is formed by assigning an orientation to each edge of a connected and acyclic undirected graph.

A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic.

A polytree is an example of an oriented graph.

The term polytree was coined in 1987 by Rebane and Pearl.

  • An arborescence is a directed rooted tree, i.e. a directed acyclic graph in which there exists a single source node that has a unique path to every other node. Every arborescence is a polytree, but not every polytree is an arborescence.
  • A multitree is a directed acyclic graph in which the subgraph reachable from any node forms a tree. Every polytree is a multitree.
  • The reachability relationship among the nodes of a polytree forms a partial order that has order dimension at most three. If the order dimension is three, there must exist a subset of seven elements <math>x</math>, <math>y_i</math>, and <math>z_i</math> such that, for either <math>x\le y_i\ge z_i</math> or <math>x\ge y_i\le z_i</math>, with these six inequalities defining the polytree structure on these seven elements.
  • A fence or zigzag poset is a special case of a polytree in which the underlying tree is a path and the edges have orientations that alternate along the path. The reachability ordering in a polytree has also been called a generalized fence.

Enumeration

The number of distinct polytrees on <math>n</math> unlabeled nodes, for <math>n=1,2,3,\dots</math>, is

Sumner's conjecture

Sumner's conjecture, named after David Sumner, states that tournaments are universal graphs for polytrees, in the sense that every tournament with <math>2n-2</math> vertices contains every polytree with <math>n</math> vertices as a subgraph. Although it remains unsolved, it has been proven for all sufficiently large values of <math>n</math>.

Applications

Polytrees have been used as a graphical model for probabilistic reasoning. If a Bayesian network has the structure of a polytree, then belief propagation may be used to perform inference efficiently on it.