# Multitree

In combinatorics and order-theoretic mathematics, a multitree may describe either of two equivalent structures: a directed acyclic graph (DAG) in which the set of vertices reachable from any vertex induces a tree, or a partially ordered set (poset) that does not have four items a, b, c, and d forming a diamond suborder with abd and acd but with b and c incomparable to each other (also called a diamond-free poset).

In computational complexity theory, multitrees have also been called strongly unambiguous graphs or mangroves; they can be used to model nondeterministic algorithms in which there is at most one computational path connecting any two states.

Multitrees may be used to represent multiple overlapping taxonomies over the same ground set. If a family tree may contain multiple marriages from one family to another, but does not contain marriages between any two blood relatives, then it forms a multitree.

## Equivalence between DAG and poset definitions

In a directed acyclic graph, if the set of vertices reachable from any vertex induces a tree, or equivalently if there is at most one directed path between any two vertices in either direction, then its reachability relation is a diamond-free partial order. Conversely, in a partial order, if it is diamond-free, then its transitive reduction identify a directed acyclic graph in which the set of vertices reachable from any vertex induces a tree

## Diamond-free families

A diamond-free family of sets is a family F of sets whose inclusion ordering forms a diamond-free poset. If D(n) denotes the largest possible diamond-free family of subsets of an n-element set, then it is known that

$2\leq \lim _{n\to \infty }D(n){\Big /}{\binom {n}{\lfloor n/2\rfloor }}\leq 2{\frac {3}{11}}$ and it is conjectured that the limit is 2.

A polytree, a directed acyclic graph formed by assigning an orientation to each edge of an undirected tree, may be viewed as a special case of a multitree.

The set of all vertices connected to any vertex in a multitree forms an arborescence.

The word "multitree" has also been used to refer to a series-parallel partial order, or to other structures formed by combining multiple trees.