  The Erd\H{o}s-S\'os conjecture states that the maximum number of edges in an
$n$-vertex graph without a given $k$-vertex tree is at most $\frac
{n(k-2)}{2}$. Despite significant interest, the conjecture remains unsolved.
Recently, Caro, Patk\'os, and Tuza considered this problem for host graphs that
are connected. Settling a problem posed by them, for a $k$-vertex tree $T$, we
construct $n$-vertex connected graphs that are $T$-free with at least
$(1/4-o_k(1))nk$ edges, showing that the additional connectivity condition can
reduce the maximum size by at most a factor of 2. Furthermore, we show that
this is optimal: there is a family of $k$-vertex brooms $T$ such that the
maximum size of an $n$-vertex connected $T$-free graph is at most
$(1/4+o_k(1))nk$.
