  The concept of pedant tree-connectivity was introduced by Hager in 1985. For
a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of at least two vertices,
\emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply,
\emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with
$S\subseteq V'$. For an $S$-Steiner tree, if the degree of each vertex in $S$
is equal to one, then this tree is called a \emph{pedant $S$-Steiner tree}. Two
pedant $S$-Steiner trees $T$ and $T'$ are said to be \emph{internally disjoint}
if $E(T)\cap E(T')=\varnothing$ and $V(T)\cap V(T')=S$. For $S\subseteq V(G)$
and $|S|\geq 2$, the \emph{local pedant-tree connectivity} $\tau_G(S)$ is the
maximum number of internally disjoint pedant $S$-Steiner trees in $G$. For an
integer $k$ with $2\leq k\leq n$, \emph{$k$-pedant tree-connectivity} is
defined as $\tau_k(G)=\min\{\tau_G(S)\,|\,S\subseteq V(G),|S|=k\}$. In this
paper, we first study the sharp bounds of pedant tree-connectivity. Next, we
obtain the exact value of a threshold graph, and give an upper bound of the
pedant-tree $k$-connectivity of a complete multipartite graph. For a connected
graph $G$, we show that $0\leq \tau_k(G)\leq n-k$, and graphs with
$\tau_k(G)=n-k,n-k-1,n-k-2,0$ are characterized in this paper. In the end, we
obtain the Nordhaus-Guddum type results for pedant tree-connectivity.
