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We consider the exact multiplicity of nodal solutions of the boundary value problem $$displaylines{ u''+lambda f(u)=0 , quad tin (0, 1),cr u'(0)=0,quad u(1)=0, }$$ where $lambda in mathbb{R}$ is a positive parameter. $fin C^1(mathbb{R}, mathbb{R})$ satisfies $f'(u)>frac{f(u)}{u}$, if $u eq 0$. There exist $heta_1<s_1<0<s_2<heta_2$ such that $f(s_1)=f(0)=f(s_2)=0$; $uf(u)>0$, if $u<s_1$ or $u>s_2$; $uf(u)<0$, if $s_1<u<s_2$ and $u eq 0$; $int_{heta_1}^0 f(u)du=int_0^{heta_2} f(u)du=0$. The limit $f_infty=lim_{so infty} frac{f(s)}{s}in (0,infty)$. Using bifurcation techniques and the Sturm comparison theorem, we obtain curves of solutions which bifurcate from infinity at the eigenvalues of the corresponding linear problem, and obtain the exact multiplicity of solutions to the problem for $lambda$ lying in some interval in $mathbb{R}$.