  We initiate a systematic study of the class of theories without the tree
property of the second kind - NTP2. Most importantly, we show: the burden is
"sub-multiplicative" in arbitrary theories (in particular, if a theory has TP2
then there is a formula with a single variable witnessing this); NTP2 is
equivalent to the generalized Kim's lemma and to the boundedness of ist-weight;
the dp-rank of a type in an arbitrary theory is witnessed by mutually
indiscernible sequences of realizations of the type, after adding some
parameters - so the dp-rank of a 1-type in any theory is always witnessed by
sequences of singletons; in NTP2 theories, simple types are co-simple,
characterized by the co-independence theorem, and forking between the
realizations of a simple type and arbitrary elements satisfies full symmetry; a
Henselian valued field of characteristic (0,0) is NTP2 (strong, of finite
burden) if and only if the residue field is NTP2 (the residue field and the
value group are strong, of finite burden respectively), so in particular any
ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2
theory preserves NTP2.
