  We consider the ferromagnetic Ising model on a sequence of graphs $G_n$
converging locally weakly to a rooted random tree. Generalizing [Montanari,
Mossel, Sly '11], under an appropriate "continuity" property, we show that the
Ising measures on these graphs converge locally weakly to a measure, which is
obtained by first picking a random tree, and then the symmetric mixture of
Ising measures with $+$ and $-$ boundary conditions on that tree. Under the
extra assumptions that $G_n$ are edge-expanders, we show that the local weak
limit of the Ising measures conditioned on positive magnetization, is the Ising
measure with $+$ boundary condition on the limiting tree. The "continuity"
property holds except possibly for countably many choices of $\beta$, which for
limiting trees of minimum degree at least three, are all within certain
explicitly specified compact interval. We further show the edge-expander
property for (most of) the configuration model graphs corresponding to limiting
(multi-type) Galton Watson trees.
