Consider the  Dirichlet problem of   a  semilinear reaction  diffusion
equation  $u_t =  \Delta u +  f(t,u)$  on an $N$-dimensional ball with
time  periodic nonlinear  term  $f$.   We give  a  complete  long-term
description  of  spatio-temporal asymptotics  for  bounded nonnegative
solutions,  by a dynamical  systems  approach.  A substantial part  of
this work   is to establish  the zero  number diminishing property for
radial  solutions  of parabolic equations.  With  the  aid of the zero
number technique, we find  a nontrivial gradient dynamic  structure
of the radial problem.  This structure is linked with the dynamics of the
original problem  by asymptotic spatial  symmetrization of nonnegative
solutions.  Combined further with  spatial symmetry of  eigenspaces of
linearized operators and with normal hyperbolic behavior near continua
of fixed points,  we obtain our  main result  to  the effect  that any
bounded nonnegative  solution exhibits  asymptotic periodicity in time
and asymptotic radial symmetry in space as $t \to \infty$.