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@@ -131,7 +131,15 @@ with the elements of $Q$ such that
 \end{itemize}
 
 A run is \definedterm{accepting} if $T$'s root gets relabeled to an accepting
-state, that is a state $q \in F$.
+state, that is a state $q \in F$. The set of all trees accepted by an automaton
+$A$ is called the \definedterm{language recognized by $A$}, notated $L(A)$. We
+call the class of all languages recognized by tree automata \definedterm{regular
+tree languages}, analogously to regular languages recognized by finite state
+automata.
+
+We note that the expressive power of deterministic bottom-up tree automata is
+the same as that of nondeterministic (either bottom-up or top-down) tree
+automata.
 
 \subsection{Monadic Second Order (MSO) Logic}
 % def of MSO
@@ -167,6 +175,8 @@ f
 
 \chapter{Branch Infix Regular Queries}\label{r:branchinfix}
 
+In this chapter we will present a solution to the following query problem:
+
 \queryproblem[%
     regular language $L$ over alphabet $\Sigma$.
 ]{Branch Infix Regular Queries}{%