  We obtain an improved lower bound for the regularity of the binomial edge
ideals of trees. We prove an upper bound for the regularity of the binomial
edge ideals of certain subclass of block-graphs. As a consequence we obtain
sharp upper and lower bounds for the regularity of binomial edge ideals of a
class of trees called lobsters. We also obtain precise expressions for the
regularities of binomial edge ideals of certain classes of trees and block
graphs.
