Rooting T in an arbitrary node r gives rise to the rooted tree T r.
Given x a node of T, let Ψ(x; T) represent the set of trees obtained by rooting T at x and deleting x and its incident edges from T. Definition 14. Define the subrank subrk(x; T) to be max S ∈Ψ(x ; T ) rk(S), the maximum rank of the subtrees of T when rooted at x.
This method works by computing the RF distance between the rooted phylogenetic trees and S ¯ obtained by rooting T and S ¯ at any leaf label in X′∩ Y′. (Note that if X ∩ ϕ (ℒ (x ˙ ) ) = ∅ or X ∩ ϕ (ℒ (y ˙ ) ) = ∅, then 's distance from S ¯ is the same as its distance from).
Given a taxon set S and a phylogeny T on S, we can see that trimming all the other nodes (including the root if T is rooted) from T gives exactly one topology for every quartet of S. The quartet-based phylogeny reconstruction works inversely to first build a phylogeny for every quartet and then infer an overall phylogeny for the whole set of taxa.
The four topology-labeled, three-leaved, rooted trees, namely, t, t, t and t, with leaf label set {1, 2, 3}, are depicted in Figure 1(i)–(iv).
A rooted subtree is a subtree that can be obtained from a rooted tree T by removing an edge of T and taking the component that does not contain the original root of T. The proximal direction in a rooted tree is towards the root, while the distal direction is away from the root.
In the unipartite case, ε t = L tR t, being L t and R t the number of leaves in the left and right subtrees rooted at t.
It recursively splits the structure and thereby derives a binary decomposition tree rooted in T and whose leaves correspond to T-substructures.
As we consider rooted trees T only, we adopt the convention that an edge denoted (v, u) means that u < T v.
If e is an edge of a rooted tree T, we write pa e) and ch e) for its parent vertex (origin) and its child vertex (end), respectively.
