  We study the effect of boundary conditions on the relaxation time of the
Glauber dynamics for the hard-core model on the tree. The hard-core model is
defined on the set of independent sets weighted by a parameter $\lambda$,
called the activity. The Glauber dynamics is the Markov chain that updates a
randomly chosen vertex in each step. On the infinite tree with branching factor
$b$, the hard-core model can be equivalently defined as a broadcasting process
with a parameter $\omega$ which is the positive solution to
$\lambda=\omega(1+\omega)^b$, and vertices are occupied with probability
$\omega/(1+\omega)$ when their parent is unoccupied. This broadcasting process
undergoes a phase transition between the so-called reconstruction and
non-reconstruction regions at $\omega_r\approx \ln{b}/b$. Reconstruction has
been of considerable interest recently since it appears to be intimately
connected to the efficiency of local algorithms on locally tree-like graphs,
such as sparse random graphs. In this paper we show that the relaxation time of
the Glauber dynamics on regular $b$-ary trees $T_h$ of height $h$ and $n$
vertices, undergoes a phase transition around the reconstruction threshold. In
particular, we construct a boundary condition for which the relaxation time
slows down at the reconstruction threshold. More precisely, for any $\omega \le
\ln{b}/b$, for $T_h$ with any boundary condition, the relaxation time is
$\Omega(n)$ and $O(n^{1+o_b(1)})$. In contrast, above the reconstruction
threshold we show that for every $\delta>0$, for $\omega=(1+\delta)\ln{b}/b$,
the relaxation time on $T_h$ with any boundary condition is $O(n^{1+\delta +
o_b(1)})$, and we construct a boundary condition where the relaxation time is
$\Omega(n^{1+\delta/2 - o_b(1)})$.
