Apolyforest (ordirected forestororiented 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.
Anarborescence 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.
Amultitree 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 , , and (for ) such that, for each , either or, with these six inequalities defining the polytree structure on these seven elements.[6]
Afence 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.[7]
Sumner's conjecture, named after David Sumner, states that tournaments are universal graphs for polytrees, in the sense that every tournament with vertices contains every polytree with vertices as a subgraph. Although it remains unsolved, it has been proven for all sufficiently large values of .[8]
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.[9]
Harary, Frank; Sumner, David (1980), "The dichromatic number of an oriented tree", Journal of Combinatorics, Information & System Sciences, 5 (3): 184–187, MR0603363.