  We introduce and examine some special classes of invariant
types$\unicode{x2014}$bi-invariant, strongly bi-invariant, extendibly
invariant, and reliably invariant types$\unicode{x2014}$and show that they are
related to certain model-theoretic tree properties.
  We show that the comb tree property (recently introduced by Mutchnik) is
equivalent to the failure of Kim's lemma for bi-invariant types and is implied
by the failure of Kim's lemma for reliably invariant types over invariance
bases. We show that every type over an invariance base extends to a reliably
invariant type$\unicode{x2014}$generalizing an unpublished result of Kruckman
and Ramsey$\unicode{x2014}$and use this to show that, under a reasonable
definition of Kim-dividing, Kim-forking coincides with Kim-dividing over
invariance bases in theories without the comb tree property. Assuming a
measurable cardinal, we characterize the comb tree property in terms of a form
of dual local character.
  We also show that the antichain tree property (introduced by Ahn and Kim)
seems to have a somewhat similar relationship to strong bi-invariance. In
particular, we show that NATP theories satisfy Kim's lemma for strongly
bi-invariant types and (assuming a measurable cardinal) satisfy a different
form of dual local character. Furthermore, we examine a mutual generalization
of the local character properties satisfied by NTP$_2$ and NSOP$_1$ theories
and show that it is satisfied by all NATP theories.
  Finally, we give some related minor results$\unicode{x2014}$a strengthened
local character characterization of NSOP$_1$ and a characterization of coheirs
in terms of invariant extensions in expansions$\unicode{x2014}$as well as a
pathological example of Kim-dividing.
