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The concept of maximum local connectivity $\bar {\kappa}$ of a graph wasintroduced by Bollob\'{a}s. One of the problems about it is to determine thelargest number of edges $f(n;\bar{\kappa}\leq \ell)$ for graphs of order $n$that have local connectivity at most $\ell$. We consider a generalization ofthe above concept and problem. For $S\subseteq V(G)$ and $|S|\geq 2$, the\emph{generalized local connectivity} $\kappa(S)$ is the maximum number ofinternally disjoint trees connecting $S$ in $G$. The parameter$\bar{\kappa}_k(G)=max\{\kappa(S)|S\subseteq V(G),|S|=k\}$ is called the\emph{maximum generalized local connectivity} of $G$. This paper it to considerthe problem of determining the largest number $f(n;\bar{\kappa}_k\leq \ell)$ ofedges for graphs of order $n$ that have maximum generalized local connectivityat most $\ell$. The exact value of $f(n;\bar{\kappa}_k\leq \ell)$ for $k=n,n-1$is determined. For a general $k$, we construct a graph to obtain a sharp lowerbound.

On extremal graphs with at most $\ell$ internally disjoint Steiner trees connecting any n-1 vertices