We study the existence of nodal solutions of the $m$-point boundary value problem \[ u^{\prime \prime }+ f(u)=0, \quad 0<t<1, u^{\prime }(0)=0, \quad u(1)=\sum ^{m-2}_{i=1} \alpha _i u(\eta _i) \] where $\eta _i\in \mathbb{Q}$ $(i=1, 2, \cdots , m-2)$ with $0<\eta _1<\eta _2<\cdots <\eta _{m-2}<1$, and $\alpha _i\in \mathbb{R}$ $(i=1, 2, \cdots , m-2)$ with $\alpha _i>0$ and $0<\sum \nolimits ^{m-2}_{i=1} \alpha _i < 1$. We give conditions on the ratio $f(s)/s$ at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation techniques.