How connectivity affects the extremal number of trees

The Erdős-Sós 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, Patkós, 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 \((1/4+o_k(1))nk\).