Figure 9:
Generation of compact coalescent histories of lodgepole labeled topologies. (A) Generation of compact histories of λn+1 from a compact history of λn. Let h be a compact history of λn with label m = m(h) for its root branch. The compact histories h′ of λn+1 generated by h are determined by choosing two parameters: (i) the label, 0 or 1, for the branch above the cherry root subtree of λn+1, and (ii) the label ℓ ∈ [0, m] for the branch above the root subtree λn of λn+1. If the label in (i) is chosen to be 0, then the label m(h′) = m + 2 − ℓ of the root branch in λn+1 ranges in the interval m(h′) ∈ [2, m + 2]. Similarly, if the label chosen in (i) is 1, then the label m(h′) = m + 1 − ℓ ranges in m(h′) ∈ [1, m + 1]. (B) The first levels of the generating tree (Eq. 11). A node (m) at depth n in the generating tree accounts for a compact history of λn with root branch labeled by m. The root of the generating tree has label (0), as the lodgepole λ0 with 1 taxon has no coalescent events. Nodes descending from a generic node (m) are determined by Eq. 11. The 10 nodes at depth 2 account for the compact histories of λ2 of Fig. 8.