Let $f\colon[0,\infty)\rightarrow{\bf R}^+$ be convex, $b\in {\bf R}$, and let $\alpha_i$, $\alpha'_i$ and $\gamma_i$, $i=0,1$, be nonnegative numbers. Consider a two-point boundary value problem with a parameter $\lambda\colon(\ast)$ $u''+bu'+\lambda f(u)=0$, $\alpha_0u(0)-\alpha'_0u'(0)=\gamma_0$, $\alpha_1u'(1)=\gamma_1$. With essentially no additional assumptions on $f$, the author gives an almost complete description of the positive solutions of the BVP ($*$), describing in detail how the structure of the solution set depends on the asymptotic behavior of $f$. Let $\psi_\infty$ be the positive normalized eigenfunction of $L^*\varphi=\mu m_\infty\varphi$ on $(0,1)$, $B^*\varphi=0$, where $L^*\varphi=-\varphi''+b\varphi'$ and $B^*=(B^*_0,B^*_1)$ with $B^*_i\varphi=[\alpha_i+(-1)^i\alpha'_i] u(i)-(-1)^i\alpha'_iu'(i)$, $f(t)/t\rightarrow m_\infty\in [0,\infty]$ as $t\rightarrow\infty$ and $\nu=\int^1_0\psi_\infty g$ with $\nu=0$ if $m_\infty=0$, where $g$ is the unique solution of $(\ast)$ with $f\equiv0$. The author shows that there is a $\lambda^*>0$ such that, depending on monotonicity properties of $(f(u)+m_\infty\nu)/u$ and $m_\infty,\;{\rm the \;BVP}(*)$ possesses exactly one or two solutions for all $\lambda\in(0,\lambda^*)$ or $\lambda\in(0,\bar\lambda)$ for some $\bar\lambda<\lambda^*$ while, for $\lambda=\lambda^*$, either there is no solution, there is exactly one solution or an infinite number of solutions. In both cases, the behavior of the curve $\lambda\rightarrow(\lambda,u(\lambda))$ is also studied.