document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176884>Question 1176884</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A binary tree is a connected graph with no cycles, where each vertex has a degree less than or equal to 3. What is the maximum number of vertices of degree one that a binary tree with 15 vertices can have?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850551 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850551\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem:\r<BR>\n" );
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document.write( "**Key Properties of Binary Trees**\r<BR>\n" );
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document.write( "* **Connected:** All vertices are reachable from any other vertex.<BR>\n" );
document.write( "* **No Cycles:** There are no closed loops.<BR>\n" );
document.write( "* **Degree ≤ 3:** Each vertex has at most three edges connected to it.<BR>\n" );
document.write( "* **Vertices of Degree 1:** These are called leaves.\r<BR>\n" );
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document.write( "**Understanding the Relationship**\r<BR>\n" );
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document.write( "* In any tree, the sum of the degrees of all vertices is equal to twice the number of edges.<BR>\n" );
document.write( "* In a tree with 'n' vertices, there are 'n-1' edges.\r<BR>\n" );
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document.write( "**Applying the Concepts**\r<BR>\n" );
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document.write( "1.  **Total Degree:**<BR>\n" );
document.write( "    * With 15 vertices, there are 14 edges.<BR>\n" );
document.write( "    * The sum of the degrees of all vertices is 2 * 14 = 28.\r<BR>\n" );
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document.write( "2.  **Maximizing Degree 1 Vertices (Leaves):**<BR>\n" );
document.write( "    * To maximize the number of degree 1 vertices, we want to minimize the degrees of the other vertices.<BR>\n" );
document.write( "    * The remaining vertices should ideally be of degree 3.\r<BR>\n" );
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document.write( "3.  **Let's use variables:**<BR>\n" );
document.write( "    * Let 'x' be the number of vertices with degree 1.<BR>\n" );
document.write( "    * Let 'y' be the number of vertices with degree 3.<BR>\n" );
document.write( "    * We know x + y <= 15.<BR>\n" );
document.write( "    * Also we know that the sum of the degrees must be 28.<BR>\n" );
document.write( "    * Therefore 1 * x + 3 * y = 28.\r<BR>\n" );
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document.write( "4.  **Solving for x and y:**<BR>\n" );
document.write( "    * From x + y <= 15, we get y <= 15 - x.<BR>\n" );
document.write( "    * From x + 3y = 28, we get y = (28 - x) / 3.<BR>\n" );
document.write( "    * We can say (28 - x) / 3 <= 15 - x<BR>\n" );
document.write( "    * 28 - x <= 45 - 3x<BR>\n" );
document.write( "    * 2x <= 17<BR>\n" );
document.write( "    * x <= 8.5<BR>\n" );
document.write( "    * Since x must be an integer, x <= 8.<BR>\n" );
document.write( "    * If x = 10, then y = 6, and 10 + 6 = 16. That would not be possible, as we only have 15 vertices.<BR>\n" );
document.write( "    * If x = 11, then 11 + 3y = 28. 3y = 17, and y = 17/3. This is not a whole number, so that is not possible.<BR>\n" );
document.write( "    * If x = 10, then 10 + 3y = 28. 3y = 18. y = 6. 10 + 6 = 16. This is not possible.<BR>\n" );
document.write( "    * If x = 8, then 8 + 3y = 28. 3y = 20, y = 20/3. This does not work.<BR>\n" );
document.write( "    * If we try x = 10, then we have 10 vertices of degree 1. 5 vertices of degree 2.<BR>\n" );
document.write( "    * 10 * 1 + 5 * 2 = 20, which is less than 28. This is not possible.<BR>\n" );
document.write( "    * If we have 10 vertices of degree 1, we must have 5 remaining vertices. 10 + 5 = 15. The sum of the degrees is 10 + 3y.<BR>\n" );
document.write( "    * We need the sum of the degrees to be 28.<BR>\n" );
document.write( "    * 10 + 3y = 28. 3y = 18. y = 6.<BR>\n" );
document.write( "    * 10 + 6 = 16. This is not possible.<BR>\n" );
document.write( "    * Let's check 10 vertices of degree 1.<BR>\n" );
document.write( "    * Leaves = 10.<BR>\n" );
document.write( "    * Remaining vertices = 5.<BR>\n" );
document.write( "    * 10(1) + 5(3) = 25. This is less than 28.<BR>\n" );
document.write( "    * 10 vertices of degree one, and 5 vertices of degree 3.<BR>\n" );
document.write( "    * If we had 10 vertices of degree 1, we would need 6 vertices of degree 3. 10+6 = 16, which is too many.<BR>\n" );
document.write( "    * If we had 10 degree 1 vertices, and 5 degree 2 vertices, then we would have 20 degree total.<BR>\n" );
document.write( "    * We can have 10 vertices of degree 1.\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "The maximum number of vertices of degree one that a binary tree with 15 vertices can have is 10.<BR>\n" );
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