  The Connectivity Augmentation Problem (CAP) together with a well-known
special case thereof known as the Tree Augmentation Problem (TAP) are among the
most basic Network Design problems. There has been a surge of interest recently
to find approximation algorithms with guarantees below $2$ for both TAP and
CAP, culminating in the currently best approximation factor for both problems
of $1.393$ through quite sophisticated techniques.
  We present a new and arguably simple matching-based method for the well-known
special case of leaf-to-leaf instances. Combining our work with prior
techniques, we readily obtain a $(\frac{4}{3}+\epsilon)$-approximation for
Leaf-to-Leaf CAP by returning the better of our solution and one of an existing
method. Prior to our work, a $\frac{4}{3}$-guarantee was only known for
Leaf-to-Leaf TAP instances on trees of height $2$. Moreover, when combining our
technique with a recently introduced stack analysis approach, which is part of
the above-mentioned $1.393$-approximation, we can further improve the
approximation factor to $1.29$, obtaining for the first time a factor below
$\frac{4}{3}$ for a nontrivial class of TAP/CAP instances.
