  Motivated from the study of eccentricity, center, and sum of eccentricities
in graphs and trees, we introduce several new distance-based global and local
functions based on the smallest distance from a vertex to some leaf (called the
`uniformity' at that vertex). Some natural extremal problems on trees are
considered. Then the middle parts of a tree is discussed and compared with the
well-known center of a tree. The values of the global functions are also
compared with the sum of eccentricities and some sharp bounds are established.
Last but not the least, we show that the difference between the eccentricity
and the uniformity, when considered as a local function, behaves in a very
similar way as the eccentricity itself.
