  Planet formation around one component of a tight, eccentric binary system
such as $\gamma$ Cephei (with semimajor axis around 20 AU) is theoretically
challenging because of destructive high-velocity collisions between
planetesimals. Despite this fragmentation barrier, planets are known to exist
in such orbital configurations. Here we present a novel numerical framework for
carrying out multi-annulus coagulation-fragmentation calculations of
planetesimal growth, which fully accounts for the specifics of planetesimal
dynamics in binaries, details of planetesimal collision outcomes, and the
radial transport of solids in the disk due to the gas drag-driven inspiral. Our
dynamical inputs properly incorporate the gravitational effects of both the
eccentric stellar companion and the massive non-axisymmetric protoplanetary
disk in which planetesimals reside, as well as gas drag. We identify a set of
disk parameters that lead to successful planetesimal growth in systems such as
$\gamma$ Cephei or $\alpha$ Centauri starting from $1-10$ km size objects. We
identify the apsidal alignment of a protoplanetary disk with the binary orbit
as one of the critical conditions for successful planetesimal growth: It
naturally leads to the emergence of a dynamically quiet location in the disk
(as long as the disk eccentricity is of order several percent), where favorable
conditions for planetesimal growth exist. Accounting for the gravitational
effect of a protoplanetary disk plays a key role in arriving at this
conclusion, in agreement with our previous results. These findings lend support
to the streaming instability as the mechanism of planetesimal formation. They
provide important insights for theories of planet formation around both binary
and single stars, as well as for the hydrodynamic simulations of protoplanetary
disks in binaries (for which we identify a set of key diagnostics to verify).
