  We present in this work a study of tree-dominated charmless three-body decays
of $B$ mesons, $B^-\to K^+K^-\pi^-$ and $B^-\to\pi^+\pi^-\pi^-$, within the
factorization approach. The main results are: (i) There are two distinct
sources of nonresonant contributions: one arises from the $b\to u$ tree
transition and the other from the nonresonant matrix element of scalar
densities $\langle M_1M_2|\bar q_1 q_2|0\rangle^{\rm NR}$. It turns out that
even for tree-dominated three-body decays, dominant nonresonant contributions
originate from the penguin diagram rather than from the $b\to u$ tree process,
as implied by the large nonresonant component observed recently in the $\pi^-
K^+$ system which accounts for one third of the $B^-\to K^+K^-\pi^-$ rate. (ii)
The calculated branching fraction of $B^-\to f_2(1270)\pi^-\to K^+K^-\pi^-$ is
smaller than the LHCb by a factor of $\sim 7$ in its central value, but the
predicted $\B(B^-\to f_2(1270)\pi^-\to\pi^+\pi^-\pi^-)$ is consistent with the
data. Branching fractions of $B^-\to f_2(1270)\pi^-$ extracted from the LHCb
measurements of these two processes also differ by a factor of seven!
Therefore, it is likely that the $f_2(1270)$ contribution to $B^-\to
K^+K^-\pi^-$ is largely overestimated experimentally. Including $1/m_b$ power
corrections from penguin annihilation inferred from QCD factorization (QCDF), a
sizable CP asymmetry of 32\% in the $f_2(1270)$ component agrees with
experiment. (iii) A fraction of 5\% for the $\rho(1450)$ component in
$B^-\to\pi^+\pi^-\pi^-$ is in accordance with the theoretical expectation.
However, a large fraction of 30\% in $B^-\to K^+K^-\pi^-$ is entirely
unexpected. This issue needs to be clarified in the future.
