  An $r$-uniform hypergraph is a tight $r$-tree if its edges can be ordered so
that every edge $e$ contains a vertex $v$ that does not belong to any preceding
edge and the set $e-v$ lies in some preceding edge. A conjecture of Kalai
[Kalai], generalizing the Erd\H{o}s-S\'os Conjecture for trees, asserts that if
$T$ is a tight $r$-tree with $t$ edges and $G$ is an $n$-vertex $r$-uniform
hypergraph containing no copy of $T$ then $G$ has at most
$\frac{t-1}{r}\binom{n}{r-1}$ edges.
  A trunk $T'$ of a tight $r$-tree $T$ is a tight subtree such that every edge
of $T-T'$ has $r-1$ vertices in some edge of $T'$ and a vertex outside $T'$.
For $r\ge 3$, the only nontrivial family of tight $r$-trees for which this
conjecture has been proved is the family of $r$-trees with trunk size one in
[FF] from 1987. Our main result is an asymptotic version of Kalai's conjecture
for all tight trees $T$ of bounded trunk size. This follows from our upper
bound on the size of a $T$-free $r$-uniform hypergraph $G$ in terms of the size
of its shadow. We also give a short proof of Kalai's conjecture for tight
$r$-trees with at most four edges. In particular, for $3$-uniform hypergraphs,
our result on the tight path of length $4$ implies the intersection shadow
theorem of Katona [Katona].
