  We consider the state transfer properties of continuous time quantum walks on
trees of diameter 4. We characterize all pairs of strongly cospectral vertices
in trees of diameter 4, finding that they fall into pairs of three different
types. For each type, we construct an infinite family of diameter 4 trees for
which there is pretty good state transfer between the pair of strongly
cospectral vertices. Moreover, for two of those types, for each tree in the
infinite family, we give an explicit sequence of readout times at which the
fidelity of state transfer converges to $1$. For strongly cospectral vertices
of the remaining type, we identify a sequence of trees and explicit readout
times so that the fidelity of state transfer between the strongly cospectral
vertices approaches $1.$
  We also prove a result of independent interest: for a graph with the property
that the fidelity of state transfer between a pair of vertices at time $t_k$
converges to $1$ as $k \rightarrow \infty,$ then the derivative of the fidelity
at $t_k$ converges to $0$ as $k \rightarrow \infty. $
