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.

The number of distinct polytrees on ${\displaystyle n}$ unlabeled nodes, for ${\displaystyle n=1,2,3,\dots }$, is

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

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.

The contour tree of a real-valued function on a vector space is a polytree that describes the level sets of the function. The nodes of the contour tree are the level sets that pass through a critical point of the function and the edges describe contiguous sets of level sets without a critical point. The orientation of an edge is determined by the comparison between the function values on the corresponding two level sets.