  The eccentricity of a vertex, $ecc_T(v) = \max_{u\in T} d_T(v,u)$, was one of
the first, distance-based, tree invariants studied. The total eccentricity of a
tree, $Ecc(T)$, is the sum of eccentricities of its vertices. We determine
extremal values and characterize extremal tree structures for the ratios
$Ecc(T)/ecc_T(u)$, $Ecc(T)/ecc_T(v)$, $ecc_T(u)/ecc_T(v)$, and
$ecc_T(u)/ecc_T(w)$ where $u,w$ are leaves of $T$ and $v$ is in the center of
$T$.
  In addition, we determine the tree structures that minimize and maximize
total eccentricity among trees with a given degree sequence.

Combinatorics