Median quartet tree search algorithms using optimal subtree prune and regraft

Table 2 HDT components types

Type	Rule	Description
I	initial	Corresponds to an internal node in R. Every I component is a leaf in the HDT.
L	initial	Corresponds to a leaf node in R. Every L component is a leaf of the HDT and is considered a C type as well.
G		Corresponds to a subtree of R (i.e., every node of R descendent from any node in a G is also in that component.)
	\(GG\rightarrow G\)	Two G components can merge if their roots are siblings in R.
	\(C\rightarrow G\)	A C can convert to a G if it corresponds to a subtree of R.
C		Corresponds to a connected subset of nodes in R. The children of a C satisfy this: there exists an R node in one child that is the ancestor of an R node in the other child.
	\(IG\rightarrow C\)	An I and a G component can merge when the root of G is a child of the internal node in R represented by I.
	\(CC\rightarrow C\)	Two C components can merge if the root of one in R is the ancestor of the other in R and they form a connected subset of nodes.

Each component in the HDT corresponds to a set of connected nodes in the reference tree R, and can have four types. To construct HDT, B13 follows the rules provided here to compose new components and transform existing ones. B13 show that these rules applied with the appropriate order result in a locally balanced hierarchy.

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