Rerooted conditioned multi-type Galton-Watson trees converge locally to multi-type sin-trees, with different limits according to which types appear infinitely often on the backwards-growing spine.
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Rerooting multi-type branching trees: the infinite spine case
Rerooted conditioned multi-type Galton-Watson trees converge locally to multi-type sin-trees, with different limits according to which types appear infinitely often on the backwards-growing spine.