Refined Buneman Trees

time in the algorithm. In [BFÖ+ 03], an algorithm is given which solves the
problem in Lemma 4 in linear time. It this text, this algorithm is called the
DISCARD-RIGHT algorithm.
The second important lemma regarding refined Buneman trees is the foundation
for the incremental algorithm presented in the article by Brodal et al.
([BFÖ+ 03]), and which is presented later in this work. It is due to Bryant and
Moulton ([BM99], proposition 3). It says that a split σ ∈ RB(δ| Xk )iseither
amemberofB xk (δ| Xk )orRB(δ| Xk−1 ). If we turn it around we can say that
given the refined Buneman tree for X k , we can calculate the refined Buneman
tree for X k+1 by looking only at splits in B xk+1 (δ| Xk+1 )andRB(δ| Xk )(with
the discussion from the previous paragraph in mind, this would be “bootstrap
set” and “candidate set”, respectively).
Lemma 5. Suppose |X| > 4, and fix x ∈ X. Ifσ = U|V is a split in RB(δ) with
x ∈ U, and|U| > 2, then either U|V ∈ B x (δ) or U −{x}|V ∈ RB(δ | X−{x} ),
or both.
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