Page 193 - Applied Probability

9. Descent Graph Methods
178

AB CD
E F
GH

1/2 1/2

1/2 1/2 1/3 2/4
(a) Labeled Descent Trees

E C F D

B A H G
(b) Founder Tree Graph
FIGURE 9.2. Construction of a Founder Tree Graph

a descent tree intersects no one typed at its associated locus, the descent
tree is isolated from all other descent trees in the founder tree graph.
As suggested above, if we assign an allele to each descent tree via the
founder gene at its root, then the fates of two connected descent trees are
coupled by the common, typed people through which they pass. For exam-
ple, if both descent trees pass through an individual having heterozygous
genotype a i /a j , then one of the descent trees must carry allele a i and the
other allele a j . They cannot produce a legal descent state if they both carry
allele a i or allele a j , or one descent tree carries a completely different al-
lele. Refinement of these simple ideas involving the founder tree graph will

permit us to compute the prior sum Prior(G) associated with a
G à 0 G∩M
descent graph.
One can subdivide the nodes of the founder tree graph into connected
components. These components are sets of descent trees and should not
be confused with the components of the descent graph, which are single
descent trees. In the founder tree graph, two nodes belong to the same
component if and only if one can travel from one node to the other by a
finite sequence of edges. A component is said to be singleton if it consists