Family Trees for Enumeration

In this paper we design efficient algorithms for enumerating (1) the rooted trees with exactly [Formula: see text] vertices, (2) the maximal planar graphs with exactly [Formula: see text] vertices and (3) the linear extensions of a given poset. Those algorithms are based on tree structures of objects, called the family trees. The first algorithm enumerates each ordered tree with exactly [Formula: see text] vertices in [Formula: see text] time for each after [Formula: see text] time preprocessing. The second algorithm enumerates each maximal planar graph with exactly [Formula: see text] vertices in [Formula: see text] time for each on average. The third algorithm enumerates each linear extension of a given poset in [Formula: see text] time for each after [Formula: see text] time preprocessing, where [Formula: see text] is the number of the element in the set of the poset.