  As a variant of the much studied Tur\'an number, $ex(n,F)$, the largest
number of edges that an $n$-vertex $F$-free graph may contain, we introduce the
connected Tur\'an number $ex_c(n,F)$, the largest number of edges that an
$n$-vertex connected $F$-free graph may contain. We focus on the case where the
forbidden graph is a tree. The celebrated conjecture of Erd\H{o}s and S\'os
states that for any tree $T$, we have $ex(n,T)\le(|T|-2)\frac{n}{2}$. We
address the problem how much smaller $ex_c(n,T)$ can be, what is the smallest
possible ratio of $ex_c(n,T)$ and $(|T|-2)\frac{n}{2}$ as $|T|$ grows. We also
determine the exact value of $ex_c(n,T)$ for small trees, in particular for all
trees with at most six vertices. We introduce general constructions of
connected $T$-free graphs based on graph parameters as longest path, matching
number, branching number, etc.
