diff --git a/chapter15.tex b/chapter15.tex
index 37bb3c1..27f44df 100644
--- a/chapter15.tex
+++ b/chapter15.tex
@@ -29,7 +29,7 @@ For example, consider the following graph:
 \path[draw,thick,-] (3) -- node[font=\small,label=left:3] {} (6);
 \end{tikzpicture}
 \end{center}
-A possible spanning tree for the graph is as follows:
+One spanning tree for the graph is as follows:
 \begin{center}
 \begin{tikzpicture}[scale=0.9]
 \node[draw, circle] (1) at (1.5,2) {$1$};
@@ -67,12 +67,9 @@ is 20, and such a tree can be constructed as follows:
 \node[draw, circle] (6) at (5,1) {$6$};
 
 \path[draw,thick,-] (1) -- node[font=\small,label=above:3] {} (2);
-%\path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (3);
-%\path[draw,thick,-] (3) -- node[font=\small,label=above:9] {} (4);
 \path[draw,thick,-] (1) -- node[font=\small,label=below:5] {} (5);
 \path[draw,thick,-] (5) -- node[font=\small,label=below:2] {} (6);
 \path[draw,thick,-] (6) -- node[font=\small,label=below:7] {} (4);
-%\path[draw,thick,-] (2) -- node[font=\small,label=left:6] {} (5);
 \path[draw,thick,-] (3) -- node[font=\small,label=left:3] {} (6);
 \end{tikzpicture}
 \end{center}
@@ -92,14 +89,11 @@ example graph is 32:
 \node[draw, circle] (4) at (6.5,2) {$4$};
 \node[draw, circle] (5) at (3,1) {$5$};
 \node[draw, circle] (6) at (5,1) {$6$};
-%\path[draw,thick,-] (1) -- node[font=\small,label=above:3] {} (2);
 \path[draw,thick,-] (2) -- node[font=\small,label=above:5] {} (3);
 \path[draw,thick,-] (3) -- node[font=\small,label=above:9] {} (4);
 \path[draw,thick,-] (1) -- node[font=\small,label=below:5] {} (5);
-%\path[draw,thick,-] (5) -- node[font=\small,label=below:2] {} (6);
 \path[draw,thick,-] (6) -- node[font=\small,label=below:7] {} (4);
 \path[draw,thick,-] (2) -- node[font=\small,label=left:6] {} (5);
-%\path[draw,thick,-] (3) -- node[font=\small,label=left:3] {} (6);
 \end{tikzpicture}
 \end{center}
 
@@ -107,15 +101,14 @@ Note that a graph may have several
 minimum and maximum spanning trees,
 so the trees are not unique.
 
-This chapter discusses algorithms
-for constructing spanning trees.
-It turns out that it is easy to find
-minimum and maximum spanning trees,
-because many greedy methods produce optimals solutions.
-We will learn two algorithms that both process
+It turns out that several greedy methods
+can be used to construct minimum and maximum
+spanning trees.
+In this chapter, we discuss two algorithms
+that process
 the edges of the graph ordered by their weights.
-We will focus on finding minimum spanning trees,
-but similar algorithms can be used for finding
+We focus on finding minimum spanning trees,
+but the same algorithms can find
 maximum spanning trees by processing the edges in reverse order.
 
 \section{Kruskal's algorithm}