  The total reciprocal edge-eccentricity is a novel graph invariant with vast
potential in structure activity/property relationships. This graph invariant
displays high discriminating power with respect to both biological activity and
physical properties. If $G=(V_G,E_G)$ is a simple connected graph, then the
total reciprocal edge-eccentricity (REE) of $G$ is defined as
$\xi^{ee}(G)=\sum_{uv\in E_G}(1/\varepsilon_G(u)+1/\varepsilon_G(v))$, where
$\varepsilon_G(v)$ is the eccentricity of the vertex $v$. In this paper we
first introduced four edge-grafting transformations to study the mathematical
properties of the reciprocal edge-eccentricity of $G$. Using these elegant
mathematical properties, we characterize the extremal graphs among $n$-vertex
trees with given graphic parameters, such as pendants, matching number,
domination number, diameter, vertex bipartition, et al. Some sharp bounds on
the reciprocal edge-eccentricity of trees are determined.
