Let $G$ be a finite connected graph with minimum degree $\delta $. The leaf number $L(G)$ of $G$ is defined as the maximum number of leaf vertices contained in a spanning tree of $G$. We prove that if $\delta \ge \frac {1}{2}(L(G)+1)$, then $G$ is 2-connected. Further, we deduce, for graphs of girth greater than 4, that if $\delta \ge \smash {\frac {1}{2}}(L(G)+1)$, then $G$ contains a spanning path. This provides a partial solution to a conjecture of the computer program Graffiti.pc [DeLaVi na and Waller, Spanning trees with many leaves and average distance, Electron. J. Combin. 15 (2008), 1–16]. For $G$ claw-free, we show that if $\delta \ge \frac {1}{2}(L(G)+1)$, then $G$ is Hamiltonian. This again confirms, and even improves, the conjecture of Graffiti.pc for this class of graphs.