  A \emph{vertex separator} of a connected graph $G$ is a set of vertices
removing which will result in two or more connected components and a
\emph{minimum vertex separator} is a set which contains the minimum number of
such vertices, i.e., the cardinality of this set is least among all possible
vertex separator sets. The cardinality of the minimum vertex separator refers
to the connectivity of the graph G. A connected graph is said to be
$k-connected$ if removing exactly $k$ vertices, $ k\geq 1$, from the graph,
will result in two or more connected components and on removing any $(k-1)$
vertices, the graph is still connected. A \emph{connectivity augmentation} set
is a set of edges which when augmented to a $k$-connected graph $G$ will
increase the connectivity of $G$ by $r$, $r \geq 1$, making the graph
$(k+r)$-$connected$ and a \emph{minimum connectivity augmentation} set is such
a set which contains a minimum number of edges required to increase the
connectivity by $r$. In this paper, we shall investigate a $r$-$connectivity$
augmentation in trees, $r \geq 2$. As part of lower bound study, we show that
any minimum $r$-connectivity augmentation set in trees requires at least $
\lceil\frac{1}{2} \sum\limits_{i=1}^{r-1} (r-i) \times l_{i} \rceil $ edges,
where $l_i$ is the number of vertices with degree $i$. Further, we shall
present an algorithm that will augment a minimum number of edges to make a tree
$(k+r)$-connected.
