  Recent work has highlighted deep connections between sequence-length
requirements for high-probability phylogeny reconstruction and the related
problem of the estimation of ancestral sequences. In [Daskalakis et al.'09],
building on the work of [Mossel'04], a tight sequence-length requirement was
obtained for the CFN model. In particular the required sequence length for
high-probability reconstruction was shown to undergo a sharp transition (from
$O(\log n)$ to $\hbox{poly}(n)$, where $n$ is the number of leaves) at the
"critical" branch length $\critmlq$ (if it exists) of the ancestral
reconstruction problem.
  Here we consider the GTR model. For this model, recent results of [Roch'09]
show that the tree can be accurately reconstructed with sequences of length
$O(\log(n))$ when the branch lengths are below $\critksq$, known as the
Kesten-Stigum (KS) bound. Although for the CFN model $\critmlq = \critksq$, it
is known that for the more general GTR models one has $\critmlq \geq \critksq$
with a strict inequality in many cases. Here, we show that this phenomenon also
holds for phylogenetic reconstruction by exhibiting a family of symmetric
models $Q$ and a phylogenetic reconstruction algorithm which recovers the tree
from $O(\log n)$-length sequences for some branch lengths in the range
$(\critksq,\critmlq)$. Second we prove that phylogenetic reconstruction under
GTR models requires a polynomial sequence-length for branch lengths above
$\critmlq$.
