Mangrove

Mangroves are various large and extensive types of trees up to medium height and shrubs that grow in saline coastal sediment habitats in the tropics and subtropics—mainly between latitudes 25° N and 25° S. The remaining mangrove forest areas of the world in 2000 was 53,190 square miles (137,760 km²) spanning 118 countries and territories.

Mangroves are salt tolerant trees, also called halophytes, and are adapted to life in harsh coastal conditions. They contain a complex salt filtration system and complex root system to cope with salt water immersion and wave action. They are adapted to the low oxygen (anoxic) conditions of waterlogged mud.

The word is used in at least three senses: (1) most broadly to refer to the habitat and entire plant assemblage or mangal, for which the terms mangrove forest biome, mangrove swamp and mangrove forest are also used, (2) to refer to all trees and large shrubs in the mangrove swamp, and (3) narrowly to refer to the mangrove family of plants, the Rhizophoraceae, or even more specifically just to mangrove trees of the genus Rhizophora.

Multitree

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

Equivalence between directed acyclic graph and poset definitions

If G is a directed acyclic graph ("DAG") in which the nodes reachable from each vertex form a tree (or equivalently, if G is a directed graph in which there is at most one directed path between any two nodes, in either direction) then the reachability relation in G forms a diamond-free partial order. Conversely, if P is a diamond-free partial order, its transitive reduction forms a DAG in which the successors of any node form 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