Let $T_1, \: \ldots, \: T_k$ subtrees of a tree $T$ such that for all $1\leq i <j \leq k $ the trees $T_i$ and $T_j$ have a vertex in common.
For all $1 \leq i < j \leq k$, the subtrees $T_i$ and $T_j$ have a vertex in common. Show that $T$ has a vertex which is in all of the $T_i$.