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Building on early work by Stevo Todorcevic, we describe a theory ofstationary subtrees of trees of successor-cardinal height. We define thediagonal union of subsets of a tree, as well as normal ideals on a tree, and wecharacterize arbitrary subsets of a non-special tree as being either stationaryor 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 anyordinal such that $2^{\left|\xi\right|} < \kappa$, and let $k$ be any naturalnumber. Then \[ \text{non-$\left(2^{<\kappa}\right)$-special tree } \to\left(\kappa + \xi \right)^2_k. \]  This is a generalization to trees of the BalancedBaumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the aboveto the cardinal $(2^{<\kappa})^+$, the simplest example of anon-$(2^{<\kappa})$-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 anyordinal such that $2^{\left|\xi\right|} < \kappa$, and let $k$ be any naturalnumber. Let $P$ be a partially ordered set such that $P \to(2^{<\kappa})^1_{2^{<\kappa}}$. Then \[ P \to \left(\kappa + \xi \right)^2_k.\]

A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees