  Building on early work by Stevo Todorcevic, we describe a theory of
stationary subtrees of trees of successor-cardinal height. We define the
diagonal union of subsets of a tree, as well as normal ideals on a tree, and we
characterize arbitrary subsets of a non-special tree as being either stationary
or non-stationary.
  We then use this theory to prove the following partition relation for trees:
  Main Theorem: Let $\kappa$ be any infinite regular cardinal, let $\xi$ be any
ordinal such that $2^{\left|\xi\right|} < \kappa$, and let $k$ be any natural
number. Then \[ \text{non-$\left(2^{<\kappa}\right)$-special tree } \to
\left(\kappa + \xi \right)^2_k. \]
  This is a generalization to trees of the Balanced
Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above
to the cardinal $(2^{<\kappa})^+$, the simplest example of a
non-$(2^{<\kappa})$-special tree.
  As a corollary, we obtain a general result for partially ordered sets:
  Theorem: Let $\kappa$ be any infinite regular cardinal, let $\xi$ be any
ordinal such that $2^{\left|\xi\right|} < \kappa$, and let $k$ be any natural
number. Let $P$ be a partially ordered set such that $P \to
(2^{<\kappa})^1_{2^{<\kappa}}$. Then \[ P \to \left(\kappa + \xi \right)^2_k.
\]
