# Polytree

In mathematics, and more specifically in graph theory, a **polytree**[1] (also called **directed tree**,[2] **oriented tree**[3][4] or **singly connected network**[5]) is a directed acyclic graph whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic.

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.[6]

## Related structures

Every arborescence (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) is a polytree, but not every polytree is an arborescence. Every polytree is a multitree, a directed acyclic graph in which the subgraph reachable from any node forms a tree.

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 *x*, *y _{i}*, and

*z*(for

_{i}*i*= 0, 1, 2) such that, for each

*i*, either

*x*≤

*y*≥

_{i}*z*, or

_{i}*x*≥

*y*≤

_{i}*z*, with these six inequalities defining the polytree structure on these seven elements.[7]

_{i}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*.[8]

## Enumeration

The number of distinct polytrees on *n* unlabeled nodes, for *n* = 1, 2, 3, ..., 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 2*n* − 2 vertices contains every polytree with *n* vertices as a subgraph. Although it remains unsolved, it has been proven for all sufficiently large values of *n*.[9]

## Applications

Polytrees have been used as a graphical model for probabilistic reasoning.[1] If a Bayesian network has the structure of a polytree, then belief propagation may be used to perform inference efficiently on it.[5][6]

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.[10]

## See also

## Notes

- Dasgupta (1999).
- Deo 1974, p. 206.
- Harary & Sumner (1980).
- Simion (1991).
- Kim & Pearl (1983).
- Rebane & Pearl (1987).
- Trotter & Moore (1977).
- Ruskey, Frank (1989), "Transposition generation of alternating permutations",
*Order*,**6**(3): 227–233, doi:10.1007/BF00563523, MR 1048093 - Kühn, Mycroft & Osthus (2011).
- Carr, Snoeyink & Axen (2000).

## References

- Carr, Hamish; Snoeyink, Jack; Axen, Ulrike (2000), "Computing contour trees in all dimensions",
*in Proc. 11th ACM-SIAM Symposium on Discrete Algorithms (SODA 2000)*, pp. 918–926 - Dasgupta, Sanjoy (1999), "Learning polytrees",
*in Proc. 15th Conference on Uncertainty in Artificial Intelligence (UAI 1999), Stockholm, Sweden, July-August 1999*(PDF), pp. 134–141. - Deo, Narsingh (1974),
*Graph Theory with Applications to Engineering and Computer Science*(PDF), Englewood, New Jersey: Prentice-Hall, ISBN 0-13-363473-6. - Harary, Frank; Sumner, David (1980), "The dichromatic number of an oriented tree",
*Journal of Combinatorics, Information & System Sciences*,**5**(3): 184–187, MR 0603363. - Kim, Jin H.; Pearl, Judea (1983), "A computational model for causal and diagnostic reasoning in inference engines",
*in Proc. 8th International Joint Conference on Artificial Intelligence (IJCAI 1983), Karlsruhe, Germany, August 1983*(PDF), pp. 190–193. - Kühn, Daniela; Mycroft, Richard; Osthus, Deryk (2011), "A proof of Sumner's universal tournament conjecture for large tournaments",
*Proceedings of the London Mathematical Society*, Third Series,**102**(4): 731–766, arXiv:1010.4430, doi:10.1112/plms/pdq035, MR 2793448. - Rebane, George; Pearl, Judea (1987), "The recovery of causal poly-trees from statistical data",
*in Proc. 3rd Annual Conference on Uncertainty in Artificial Intelligence (UAI 1987), Seattle, WA, USA, July 1987*(PDF), pp. 222–228. - Simion, Rodica (1991), "Trees with 1-factors and oriented trees",
*Discrete Mathematics*,**88**(1): 93–104, doi:10.1016/0012-365X(91)90061-6, MR 1099270. - Trotter, William T., Jr.; Moore, John I., Jr. (1977), "The dimension of planar posets",
*Journal of Combinatorial Theory, Series B*,**22**(1): 54–67, doi:10.1016/0095-8956(77)90048-X.