  We are interested in the local limits of families of random trees that
satisfy the Markov branching property, which is fulfilled by a wide range of
models. Loosely, this property entails that given the sizes of the sub-trees
above the root, these sub-trees are independent and their distributions only
depend upon their respective sizes. The laws of the elements of a Markov
branching family are characterised by a sequence of probability distributions
on the sets of integer partitions which describes how the sizes of the
sub-trees above the root are distributed.
  We prove that under some natural assumption on this sequence of
probabilities, when their sizes go to infinity, the trees converge in
distribution to an infinite tree which also satisfies the Markov branching
property. Furthermore, when this infinite tree has a single path from the root
to infinity, we give conditions to ensure its convergence in distribution under
appropriate rescaling of its distance and counting measure to a self-similar
fragmentation tree with immigration. In particular, this allows us to determine
how, in this infinite tree, the "volume" of the ball of radius $R$ centred at
the root asymptotically grows with $R$.
  Our unified approach will allow us to develop various new applications, in
particular to different models of growing trees and cut-trees, and to recover
known results. An illustrative example lies in the study of Galton-Watson
trees: the distribution of a critical Galton-Watson tree conditioned on its
size converges to that of Kesten's tree when the size grows to infinity. If
furthermore, the offspring distribution has finite variance, under adequate
rescaling, Kesten's tree converges to Aldous' self-similar CRT and the total
size of the $R$ first generations asymptotically behaves like $R^2$.
