The Erdős-Sós conjecture states that the maximum number of edges in an -vertex graph without a given -vertex tree is at most . Despite significant interest, the conjecture remains unsolved. Recently, Caro, Patkós, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them, for a -vertex tree , we construct -vertex connected graphs that are -free with at least 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 -vertex brooms such that the maximum size of an -vertex connected -free graph is at most .