  The Harary index of a graph $G$ is recently introduced topological index,
defined on the reverse distance matrix as $H(G)=\sum_{u,v \in
V(G)}\frac{1}{d(u,v)}$, where $d(u,v)$ is the length of the shortest path
between two distinct vertices $u$ and $v$. We present the partial ordering of
starlike trees based on the Harary index and we describe the trees with the
second maximal and the second minimal Harary index. In this paper, we
investigate the Harary index of trees with $k$ pendent vertices and determine
the extremal trees with maximal Harary index. We also characterize the extremal
trees with maximal Harary index with respect to the number of vertices of
degree two, matching number, independence number, radius and diameter. In
addition, we characterize the extremal trees with minimal Harary index and
given maximum degree. We concluded that in all presented classes, the trees
with maximal Harary index are exactly those trees with the minimal Wiener
index, and vice versa.

Combinatorics