[RBT] Removed some non-ASCII letters

datastruct/tree/red-black-tree/rbtree-en.tex

@@ -194,14 +194,14 @@ \subsection{Tree rotation}
 \begin{algorithmic}[1]
 \Function{Left-Rotate}{$T, x$}
   \State $p \gets$ \Call{Parent}{$x$}
-  \State $y \gets$ \Call{Right}{$x$} \Comment{Assume $y \ne NIL$}
+  \State $y \gets$ \Call{Right}{$x$} \Comment{Assume $y \ne$ NIL}
   \State $a \gets$ \Call{Left}{$x$}
   \State $b \gets$ \Call{Left}{$y$}
   \State $c \gets$ \Call{Right}{$y$}
   \State \Call{Replace}{$x, y$}
   \State \Call{Set-Children}{$x, a, b$}
   \State \Call{Set-Children}{$y, x, c$}
-  \If{$p = NIL$}
+  \If{$p = $ NIL}
     \State $T \gets y$
   \EndIf
   \State \Return $T$
@@ -211,14 +211,14 @@ \subsection{Tree rotation}
 
 \Function{Right-Rotate}{$T, y$}
   \State $p \gets$ \Call{Parent}{$y$}
-  \State $x \gets$ \Call{Left}{$y$} \Comment{Assume $x \ne NIL$}
+  \State $x \gets$ \Call{Left}{$y$} \Comment{Assume $x \ne$ NIL}
   \State $a \gets$ \Call{Left}{$x$}
   \State $b \gets$ \Call{Right}{$x$}
   \State $c \gets$ \Call{Right}{$y$}
   \State \Call{Replace}{$y, x$}
   \State \Call{Set-Children}{$y, b, c$}
   \State \Call{Set-Children}{$x, a, y$}
-  \If{$p = NIL$}
+  \If{$p = $ NIL}
     \State $T \gets x$
   \EndIf
   \State \Return $T$
@@ -228,7 +228,7 @@ \subsection{Tree rotation}
 
 \Function{Set-Left}{$x, y$}
   \State \Call{Left}{$x$} $\gets y$
-  \If{$y \ne NIL$}
+  \If{$y \ne$ NIL}
     \Call{Parent}{$y$} $\gets x$
   \EndIf
 \EndFunction
@@ -237,7 +237,7 @@ \subsection{Tree rotation}
 
 \Function{Set-Right}{$x, y$}
   \State \Call{Right}{$x$} $\gets y$
-  \If{$y \ne NIL$}
+  \If{$y \ne$ NIL}
     \Call{Parent}{$y$} $\gets x$
   \EndIf
 \EndFunction
@@ -252,16 +252,16 @@ \subsection{Tree rotation}
 \Statex
 
 \Function{Replace}{$x, y$}
-  \If{\Call{Parent}{$x$} $= NIL$}
-    \If{$y \ne NIL$}
-      \Call{Parent}{$y$} $\gets NIL$
+  \If{\Call{Parent}{$x$} = NIL}
+    \If{$y \ne$ NIL}
+      \Call{Parent}{$y$} $\gets$ NIL
     \EndIf
   \ElsIf{\textproc{Left}(\Call{Parent}{$x$}) $= x$}
     \State \textproc{Set-Left}(\Call{Parent}{$x$}, $y$)
   \Else
     \State \textproc{Set-Right}(\Call{Parent}{$x$}, $y$)
   \EndIf
-  \State \Call{Parent}{$x$} $\gets NIL$
+  \State \Call{Parent}{$x$} $\gets$ NIL
 \EndFunction
 \end{algorithmic}
 
@@ -417,7 +417,7 @@ \section{Insertion}
 \end{figure}
 
 Note that this transformation will move the redness one level up.
-So this is a bottom-up recursive fixing, the last step will make
+During the bottom-up recursive fixing, the last step will make
 the root node red. According to property 2, root is always black.
 Thus we need final fixing to revert the root color to black.
 
@@ -894,11 +894,11 @@ \section{Deletion}
 where $p 3.1$ and $p 3.2$ are two patterns as the following.
 
 \[
-p 3.1 £º \{ color(T) = \mathcal{B} \land color(T_l) = \mathcal{B}^2 \land color(T_r) = \mathcal{R} \}
+p 3.1 : \{ color(T) = \mathcal{B} \land color(T_l) = \mathcal{B}^2 \land color(T_r) = \mathcal{R} \}
 \]
 
 \[
-p 3.2 £º \{ color(T) = \mathcal{B} \land color(T_l) = \mathcal{R} \land color(T_r) = \mathcal{B}^2 \}
+p 3.2 : \{ color(T) = \mathcal{B} \land color(T_l) = \mathcal{R} \land color(T_r) = \mathcal{B}^2 \}
 \]
 
 
@@ -1134,7 +1134,7 @@ \section{More words}
 Red-black tree is the most popular implementation of balanced binary search
 tree. Another one is the AVL tree, which we'll introduce in next chapter.
 Red-black tree can be a good start point for more data structures. If we
-extend the number of children from 2 to $K$, and keep the balance as well,
+extend the number of children from 2 to $k$, and keep the balance as well,
 it leads to B-tree, If we store the data along with edge but not inside
 node, it leads to Tries. However, the multiple cases handling and the long
 program tends to make new comers think red-black tree is complex.