While @cfr provided an excellent solution to my post here, I have a problem when I insert the code in my main document, though it works fine as a standalone document. It gives me the error: ! Illegal parameter number in definition of \pgfmath@expression, suggesting the error is at \end{forest}. While forest seems to be more convenient for tree-diagrams, I find it very difficult. It will take me time to be able sort out my problem of the forest package. I am therefore reproducing my mwe in tikz, with only the tree-diagram and where I want the labels (as commented out in my code).
\documentclass[12pt]{report}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[inner sep=1.5pt]
\node (10) {z}
child { node {x}
child { node {s}}
child { node {t}}}
child { node {y}
child { node {s}}
child { node {t}}}
;
% \node (from z to x) {$\frac{\partial z}{\partial x}$};
% \node (from z to y) {$\frac{\partial z}{\partial y}$};
% \node (from x to s) {$\frac{\partial x}{\partial s}$};
% \node (from x to t) {$\frac{\partial x}{\partial t}$};
% \node (from y to s) {$\frac{\partial y}{\partial s}$};
% \node (from y to t) {$\frac{\partial y}{\partial t}$};
\end{tikzpicture}
\end{document}