  We introduce a model of branching interlacements for general critical
offspring distributions. It consists of a countable collection of infinite
tree-indexed random walk trajectories on $Z^d,d\geq5$. We show that this model
turns out to be the local limit of the tree-indexed random walk in a discrete
torus, conditioned on the size proportional to the volume of the torus. This
generalizes the previous results of Angel, R\'{a}th and the author, for the
critical geometric offspring distribution. Our model also includes the model of
random interlacements introduced by Sznitman as a degenerate case. To obtain
the local convergence, we establish results on decomposing large random trees
into small trees, local limits of random trees around a prefixed vertex, and
asymptotics of the visiting probability of a set by a tree-indexed random walk
with a given size in a torus. These auxiliary results are interesting in
themselves. As another application, we show that when $d\geq5$ the cover time
of a $d$-dimensional torus of side-length $N$ by tree-indexed random walks is
concentrated at $N^d log N^d/\mathrm{BCap}(\{0\})$.
