  Consider independent bond percolation with retention probability p on a
spherically symmetric tree Gamma. Write theta_Gamma(p) for the probability that
the root is in an infinite open cluster, and define the critical value
p_c=inf{p:theta_Gamma(p)>0}. If theta_Gamma(p_c)=0, then the root may still
percolate in the corresponding dynamical percolation process at the critical
value p_c, as demonstrated recently by Haggstrom, Peres and Steif. Here we
relate this phenomenon to the near-critical behaviour of theta_Gamma(p) by
showing that the root percolates in the dynamical percolation process if and
only if int_{p_c}^1 (theta_Gamma(p))^{-1}dp<infty. The ``only if'' direction
extends to general trees, whereas the ``if'' direction fails in this
generality.
