  We consider the tree consensus problem, an important problem in
bioinformatics. Given a rooted tree $t$ and another tree $T$, one would like to
incorporate compatible information from $T$ to $t$. This problem is a
subproblem in the tree refinement problem called the RF-Optimal Tree Refinement
Problem defined by in Christensen, Molloy, Vachaspati and Warnow [WABI'19] who
employ the greedy algorithm by Gawrychowski, Landau, Sung, and Weimann
[ICALP'18] that runs in time $O(n^{1.5}\log n)$. We give a faster algorithm for
this problem that runs in time $O(n\log n)$. Our key ingredient is a
bipartition compatibility criteria based on amortized-time leaf counters. While
this is an improvement, the fastest solution is an algorithm by Jansson, Shen,
and Sung [JACM'16] which runs in time $O(n)$.
