x1 is a 'quipyew' tree graph with special node x2, on nodes x3 (set of points; includes x2), with edges x4 (set of ordered pairs of nodes), and with other properties x5.
Such a graph is the essence of a biological family tree along (for example) the matrilineal line if one selects a special individual, reduces all other individuals in the tree according to their relation to the first while ignoring gender (so, the various other branches are folded along the ancestral line of the selected individual, which is taken to be the stem, and all overlapping nodes are made equivalent; in other words, all non-ancestral or non-self nodes of the selected indivodual, are contracted via siblinghood (all siblings are counted the same) at the leaves and then similar contractions sequentially occur up the tree); in other words, all non-self siblings are the same, all non-self parents are the same, all non-self ith cousins j-times removed are the same (for each i and j), all aunts and uncles are the same, all kth-great-grandparents are the same (for each k>1), all non-self children are the same, all lth-great-grandchildren are the same (for each l>1), etc. A quipyew tree graph (terminology invented by .krtisfranks.) is defined as follows: Define an ordered relation R on a collection of points (which will become x3) such that R is ordered, nonreflexive for any point, nonsymmetric for any pair of points, nontransitive for any pair of ordered pairs of points (this requirement is guaranteed by the next condition), and generates a tree on the nodes (with the edges being ordered pairs of points which satisfy R). For any tree-generating ordered relation S: if x S y, then x is a parent of y and y is a child of x; if x S y and y S z, then x is an ancestor of z and z is a descendant of x; "x is a non-self parent of y" means that x S y and x \neq y, and likewise for the other relations just described when modified by the descriptor "non-self". A quipyew tree T is a directed tree graph generated by the reflexive closure of R, such that T is totally connected (in the symmetric closure of R), every node in the vertex set (x3) of T has at most two non-self children and at most one non-self parent, and there exists exactly one special node x2 (usually denoted "0") in the vertex set of T with the property that for any node n in the vertex set of T, n has two non-self children only if (but not necessarily if) n is a strict ancestor of x2 according to the transitive closure of R. See also: grafmseljimca, tseingu, takni.